Equations of state for supernovae and compact stars · a. Meson-exchange models 8 b. Potential models 8 3. Interactions from chiral effective field theory and lattice QCD 9 4. Renormalization

Equations of state for supernovae and compact stars

M. Oertel*

Laboratoire Univers et Theories, CNRS, Observatoire de Paris,PSL Research University, Universite Paris Diderot, Sorbonne Paris Cite,5 place Jules Janssen, 92195 Meudon, France

M. Hempel†

Departement Physik, Universitat Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland

T. Klähn‡

Instytut Fizyki Teoretycznej, Uniwersytet Wrocławski, pl. M. Borna 9,PL-50-204 Wrocław, Poland

S. Typel§

Institut fur Kernphysik, Technische Universitat Darmstadt,Schlossgartenstraße 9, 64289 Darmstadt, Germanyand GSI Helmholtzzentrum fur Schwerionenforschung, Planckstraße 1,64291 Darmstadt, Germany

(published 15 March 2017)

A review is given of various theoretical approaches for the equation of state (EoS) of dense matter,relevant for the description of core-collapse supernovae, compact stars, and compact star mergers.The emphasis is put on models that are applicable to all of these scenarios. Such EoS models have tocover large ranges in baryon number density, temperature, and isospin asymmetry. The characteristicsof matter change dramatically within these ranges, from a mixture of nucleons, nuclei, and electronsto uniform, strongly interacting matter containing nucleons, and possibly other particles such ashyperons or quarks. As the development of an EoS requires joint efforts from many directions,different theoretical approaches are considered and relevant experimental and observationalconstraints which provide insights for future research are discussed. Finally, results from applicationsof the discussed EoS models are summarized.

DOI: 10.1103/RevModPhys.89.015007

CONTENTS

I. Introduction 2II. General Remarks on the EoS 3

A. Basic thermodynamic considerations 3B. Specific requirements for astrophysical EoSs 3

1. Equilibrium conditions 32. Charge neutrality and inhomogeneity effects 43. Range of thermodynamic variables 44. Particle degrees of freedom 6

III. Formal Approaches to the Description of Dense Matter 6A. Basic few-body interactions 7

1. Experimental data 72. Phenomenological forces 8

a. Meson-exchange models 8b. Potential models 8

3. Interactions from chiral effective field theoryand lattice QCD 9

4. Renormalization group methods and evolvedpotentials 9

B. Many-body methods for homogeneous matter 91. Ab initio methods 10

a. Self-consistent Green’s function 10b. Brueckner-Hartree-Fock 11c. Methods derived from the variational

principle 12d. Quantum Monte Carlo methods 13e. Chiral effective field theory 13f. Lattice methods 13g. Perturbative QCD 14h. Dyson-Schwinger approach 14

2. Phenomenological approaches 14a. Hadronic matter 15b. Quark matter 18

C. Clustered and nonuniform matter 191. Nuclear statistical equilibrium 19

a. Nuclear binding energies 20b. Excited states 20c. Coulomb interaction 20d. Medium modifications of heavy nuclei 20e. Cluster dissolution 20

2. Single nucleus approximation 21

*[email protected]†[email protected]‡[email protected]§[email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 89, JANUARY–MARCH 2017

0034-6861=2017=89(1)=015007(68) 015007-1 © 2017 American Physical Society

http://dx.doi.org/10.1103/RevModPhys.89.015007




3. Virial expansion 214. Quantum statistical approach 215. Generalized relativistic density functional 226. Nucleons-in-cell calculations 22

D. Phase transitions 231. Thermodynamic description of phase transitions 232. Coulomb effects 24

IV. Constraints on the EoS 25A. Terrestrial experiments 26

1. Systematics from nuclear masses and excitations 262. Nuclear resonances 26

a. Giant monopole resonance 26b. Giant dipole resonance 27c. Electric dipole polarizability 27

3. Neutron skin thicknesses 274. Heavy-ion collisions 28

B. Neutron matter calculations 30C. Astrophysical observations 31

1. Neutron star masses and radii 312. Neutron star cooling and rotation 33

D. Summary of constraints on the symmetry energy 33V. Modeling the EoS 34

A. Neutron star EoS 341. Neutron star crust EoSs and unified neutron

star EoSs 352. Composition of the neutron star core 37

B. EoS of uniform matter at finite temperature 37C. EoS of clustered matter at finite temperatures 38D. General purpose equations of state 40

1. Nucleons and nuclei as degrees of freedom 40a. H&W 40b. LS 42c. STOS 42d. FYSS 42e. HS 43f. SFHo and SFHx 43g. SHT(NL3), SHO(FSU), and SHO(FSU2.1) 43

2. Including additional degrees of freedom 433. Compatibility with experimental and

observational constraints 46VI. Applications in Astrophysics 48

A. Binaries and binary mergers 48B. Core-collapse supernovæ 49

1. Dynamics 492. PNSs, neutrino-driven winds, and nucleosynthesis 513. Black hole formation 52

VII. Summary and Conclusions 53Acknowledgments 54Appendix: Resources 54

1. EoS databases 542. Open source simulation software 54

References 55

I. INTRODUCTION

Matter under extreme conditions can be found at variousplaces in the Universe. Extremely high densities exist inneutron stars (NSs) (Glendenning, 1997; Weber, 2005;Haensel, Potekhin, and Yakovlev, 2007; Potekhin, 2010;Lattimer, 2012). Densities above nuclear saturation densityand high temperatures are reached when the core of a massivestar collapses. The resulting core-collapse supernova (CCSN)

explosion (Mezzacappa, 2005; Kotake, Sato, and Takahashi,2006; Janka et al., 2007; Ott, 2009; Janka, 2012a; Burrows,2013) leads to the formation of a protoneutron star (PNS)(Prakash et al., 1997; Pons et al., 1999) and finally to a NS ora black hole (BH). Similar densities and temperatures, buthigher isospin asymmetries (viz., a higher excess of neutronsover protons), are involved in the merging of NSs in closebinary systems, NS-NS, and NS-BH (Shibata and Taniguchi,2011; Faber and Rasio, 2012; Rosswog, 2015). The dynamicalevolution of such violent events and the structure of theemerging compact stars are determined among others by theequation of state (EoS) of matter. In addition, the EoS impactsthe conditions for nucleosynthesis and the emerging neutrinospectra. Hence, the EoS is an essential ingredient in manyastrophysical simulations.1 Many efforts are made to gain acomprehensive understanding of properties of the involvedmatter, which in several aspects are dramatically differentfrom those in terrestrial experiments.During the last decades numerous theoretical investiga-

tions, laboratory experiments, as well as astronomical obser-vations have been conducted in order to constrain thethermodynamic properties and chemical composition ofstellar matter for conditions relevant to the description ofcompact stars, CCSNe, and NS mergers; see, e.g., Klähn et al.(2006), Lattimer and Prakash (2007), and Lattimer and Lim(2013). There is an intrinsic connection between the macro-scopic structure and evolution of such astrophysical objectsand the underlying fundamental interactions between theconstituent particles at the microscopic level. This makesthe study of the aforementioned systems very rewarding asthey challenge our understanding of nature on both scales.The aim of this paper is to review existing approaches for thedescription of dense matter that can yield EoSs relevant forcompact star astrophysics from both purely theoretical andphenomenological perspectives. There is a large number ofdifferent approaches. In many cases the properties of mattercan be provided only for particular thermodynamic condi-tions, not always sufficient to describe simultaneously all theastrophysical systems we address. Hence, the main emphasisof this work is placed on the discussion of approaches to theEoS that are readily available for use in astrophysicalsimulations. Such EoSs that cover the full thermodynamicparameter range of temperature, density, and isospin asym-metry relevant for CCSNe, NSs, and compact binary mergerswe call “general purpose EoSs.”Within this review we will not consider the EoS for white

dwarfs since, although being compact stars in astronomicalterminology, the underlying microphysics is quite different.Because of the complexity of the topic we will also not discusspairing and related effects of superfluidity and superconduc-tivity; see, e.g., Lombardo and Schulze (2001), Alford et al.(2008), Chamel and Haensel (2008), and Page et al. (2014) forcorresponding reviews.

1Within this review we employ the general term “astrophysicalsimulations” or “astrophysical applications” synonymously with“astrophysical simulations of CCSNe, (proto)NSs, and compactbinary mergers involving NSs.”

M. Oertel et al.: Equations of state for supernovae and compact …

Rev. Mod. Phys., Vol. 89, No. 1, January–March 2017 015007-2

In Sec. II some basic thermodynamic considerations,definitions, and requirements for an EoS in astrophysicalapplications are discussed. In order to get an idea about thechallenge of constructing such an EoS, it is useful to state therelevant degrees of freedom and the ranges of the thermo-dynamic variables that have to be covered.It is not an easy task to obtain a reliable description of dense

matter that covers the full range of thermodynamic variables.In Sec. III formal approaches to the description of densematter are discussed. The main uncertainties arise from twosources as follows:

(1) the, at least partly, poor knowledge of the inter-actions, and

(2) the treatment of the many-body problem for stronglyinteracting particles.

Basic considerations about interactions are presented inSec. III.A. In Secs. III.B and III.C we outline currentlyavailable techniques that address the many-body problem atfinite densities and temperatures for homogeneous and inho-mogeneous matter. We briefly emphasize their respectiveadvantages and current limitations. The description of inho-mogeneous matter is particularly important for CCSNe. It isconceptually and computationally very involved and up tonow it has not been possible to apply sophisticated ab initiomany-body methods on a grand scale. Finally, Sec. III.Ddiscusses specific features that appear in the treatment ofphase transitions.The inherent uncertainties of any EoS model require a

careful analysis of and comparison with available experimen-tal and observational data. Therefore, we give an overview ofconstraints of the EoS from terrestrial experiments, theoreticalconsiderations, and astrophysical observations in Sec. IV.Section V presents an overview of EoS models for

astrophysical applications. Since EoSs for cold compact starshave been discussed extensively in the literature, only asummary of available models and their main features is given.The main emphasis is put on currently existing generalpurpose EoS.Section VI summarizes the impact of the EoS on the

astrophysics of compact stellar objects, e.g., on (proto)NSs,binary mergers, CCSNe, and the formation of BHs. The mainaim of this section is to show how different parts of the EoSand the associated uncertainties are related to potential astro-physical observations. These considerations might be usefulto identify open questions which inspire further work forimproving EoS models.The Appendix lists freely available resources, databases,

and software, which are related to the EoS and its applicationin the astrophysics of compact stellar objects.Throughout this paper we use units where kB ¼ ℏ ¼ c ¼ 1.

II. GENERAL REMARKS ON THE EoS

A. Basic thermodynamic considerations

In its most general form the expression “equation of state”is used for any relation between thermodynamic state varia-bles. Depending on the context, often we use it morespecifically for the set of thermodynamic equations that fully

specifies the state of matter under a given set of physicalconditions.EoSs are typically employed in astrophysical models that

use a hydrodynamic description of the macroscopic system. Inthis case, it is assumed that matter can be considered as a fluid,and explicit effects from the gravitational field do not have tobe included in the thermodynamic description. The construc-tion of an EoS supposes that the local system under consid-eration is in thermodynamic equilibrium. This usually meansthat intensive thermodynamic variables such as temperature,pressure, or chemical potentials are well defined and that theconditions of thermal equilibrium (equivalent to a constanttemperature throughout the chosen domain), mechanicalequilibrium (constant pressure), and chemical equilibrium(constant chemical potentials) hold. Therefore, uniformityof all independent intensive variables has to be demanded. Inorder to obtain a thermodynamically consistent approach, it ismost convenient to start from a thermodynamic potential,chosen according to the set of natural variables used, and toderive all relevant quantities by standard thermodynamicrelations; see, e.g., Landau and Lifshitz (1980) or theCompOSE manual (Klähn, Oertel, and Typel, 2013; Typel,Oertel, and Klähn, 2015). An example is the Helmholtz freeenergy FðT; fNig; VÞ [used, e.g., by Lattimer and Swesty(1991)], depending on the natural variables temperature T, theset of particle numbersNi (i ¼ 1;…; Npart), and the volume V.It attains a minimum in the ground state of the system forgiven values of the thermodynamic variables. In the thermo-dynamic limit, the actual value of V is irrelevant, and allextensive variables follow the same scaling. Therefore one canwork with ratios of extensive variable such as the particlenumber densities ni ¼ Ni=V that behave as intensive varia-bles. In general, the different particle species i are not inert butcan convert to other species by reactions. If they are inequilibrium the state of the system is characterized by anumber Ncons ≤ Npart of independent conserved charges.Thus, in general, the individual particle densities ni are notindependent, but connected by conditions of chemical equi-librium that can be expressed with the help of the particlechemical potentials μi ¼ ∂F=∂Ni. Sometimes a theoreticaldescription that starts from the grand canonical potentialΩðT; fμig; VÞ ¼ F −

PiμiNi is more convenient.

B. Specific requirements for astrophysical EoSs

1. Equilibrium conditions

An EoS can be applied only if the system is in thermo-dynamic equilibrium. In astrophysical simulations, this con-cerns, in particular, the chemical equilibrium since thermaland mechanical equilibrium are in general quickly achievedwith T and p as the associated intensive variables. Then, theuse of an EoS in chemical equilibrium is justified only ifthe time scales of the corresponding reactions are muchshorter than the time scales of the system’s hydrodynamicevolution.Chemical equilibrium among different nuclear species is

not achieved, if an ensemble of nuclei, nucleons, and electronsis considered at densities and temperatures as reached in mainsequence stars, in the outer regions of a CCSN or in explosive



nucleosynthesis. Hence the time evolution of the compositionhas to be followed with a reaction network depending on thereaction cross sections of the participating particle species.Typically it is assumed that a temperature on the order of0.5 MeVand above is sufficient to reach the so-called nuclearstatistical equilibrium (NSE) (Iliadis, 2007).A typical set of conserved charges of the system are the

total baryon number NB, the total (electric) charge numberNQ, the total electronic lepton number NLðeÞ, and the totalstrangeness number NS. The quantities Ni are thereby definedas net particle numbers. Correspondingly, for every particlethe chemical potential is given by

μi ¼ BiμB þQiμQ þ LðeÞi μLe

þ SiμS ð1Þ

with the baryon (Bi), charge (Qi), electronic lepton (LðeÞi ), and

strangeness number (Si) of the individual particle. Hence, thespecification of the baryon chemical potential μB, the chargechemical potential μQ, the electronic lepton chemical potentialμLe

, and the strangeness chemical potential μS is sufficient toobtain the chemical potential of every constituent. In particu-lar, in NSE, the chemical potential of each nucleus a withneutron number Na and proton number Za is given by

μa ¼ ðNa þ ZaÞμB þ ZaμQ ≡ Naμn þ Zaμp; ð2Þ

where μn (μp) is the chemical potential of neutrons (protons).Conditions on (electric) charge neutrality and weak equilib-rium can further reduce the number of independent particlenumbers or chemical potentials.Weak interactions, for instance the electron-capture reaction

pþ e− → nþ νe, cannot be considered in equilibrium ingeneral, since the relevant time scales can exceed thedynamical time scale of the astrophysical object of interest.In particular, in CCSNe, except at the highest densitiesroughly above nB ¼ NB=V ¼ 10−3 fm−3, no weak equilib-rium is obtained. In addition, neutrinos are not necessarily inequilibrium, neither thermal nor chemical. Usually they arenot considered within the EoS, but treated via a transportapproach. The neutrino transport equations, together with theemployed weak interaction rates, are then coupled via energy,momentum, and lepton number conservation to the hydro-dynamic evolution of the system and to the EoS. Theydetermine the electron number densities, which remain adegree of freedom of the EoS. Concerning strangenesschanging weak interactions, in the temperature and densityrange where strange particles have non-negligible abundan-ces, the time scales estimated for the relevant processes are ofthe order of 10−6 s or below; see, e.g., Brown et al. (1992).Therefore, in general strangeness changing weak equilibriumis assumed, i.e., μS ¼ 0. Hence, strangeness is not a conservedcharge and does not represent an independent thermodynamicvariable.The situation is different for the highest densities reached

in CCSNe, i.e., in hot (proto)NSs. At the prevailing hightemperatures and densities, neutrinos are trapped and equi-librium with respect to weak reactions is achieved. They canbe treated as part of the EoS, parametrized by the neutrinofraction Yνe ¼ nνe=nB or the lepton fraction YLe

¼ Yνe þ Ye

with the electron fraction Ye ¼ ne=nB. At a later cooling stageneutrinos become untrapped, i.e., their mean free pathbecomes longer than the system size and β equilibriumwithout neutrinos is established. This condition can beexpressed by setting the electronic lepton chemical potentialμLe

to zero in Eq. (1) as for cold NSs. Together with chargeneutrality it implies that ne or Ye are fixed by nB and are nolonger free variables of the EoS.Assuming lepton flavor conversion via neutrino oscillations

to be negligible, the heavy flavor lepton numbers are con-served independently of the electronic lepton number. For themoment no simulation has been performed that includesheavy charged leptons explicitly. The influence of heavyflavor leptons is expected to be small due to their high restmasses. Nevertheless, in EoSs of cold NSs muons are usuallyincluded.

2. Charge neutrality and inhomogeneity effects

In all astrophysical scenarios considered in this review, thesystem can be regarded as infinitely large on the length scalesof the microscopic model. The thermodynamic limit isreached and electric charge neutrality is required to avoidinstabilities due to the occurrence of strong electric fields.In its simplest form, charge neutrality can be formulated

as a local condition nQ ¼ PiQini ¼ 0. Thus nQ is not an

independent thermodynamic degree of freedom and it isconvenient to introduce the hadronic charge densitynq ¼

Pi0Qini, where the primed sum runs over all hadrons

(and/or quarks, if present). If electrons are the only leptoniccomponent, this implies nq ¼ ne.In the case of inhomogeneous matter, the charge distribu-

tion can be imbalanced locally. The resulting competitionbetween nuclear surface and Coulomb energies causes theformation of clusters or more complicated structures such as“pasta phases” (Ravenhall, Pethick, and Wilson, 1983;Hashimoto, Seki, and Yamada, 1984; Williams and Koonin,1985). Charge neutrality is maintained only globally. In asimple approximation, this occurrence of finite-size structureswith low and high baryon number densities can be treated as acoexistence of phases, however, surface and Coulomb effectsare neglected in this case (see Sec. III.D for details).

3. Range of thermodynamic variables

The most general case we are interested in is an EoSdepending on the temperature T, total baryon number densitynB, and the total hadronic charge densitynq or a set of equivalentthermodynamic variables. Instead of nq, the correspondingfraction Yq ¼ nq=nB may be used. The baryon number densitynB is sometimes replaced by the mass density ϱ ¼ mBnB withthe mass unit mB, which is often taken to be either the atomicmass unitmu or the neutronmassmn. As an order of magnitudeestimate, a baryon number density of 0.1 fm−3 corresponds to amass density of ϱ ≈ 1.66 × 1014 g=cm3.The observations of compact stars with masses of 2M⊙

(Demorest et al., 2010; Antoniadis et al., 2013; Fonseca et al.,2016) imply that the maximum baryon number density in NSscan approach approximately 10 times the nuclear saturationdensity nsat ≈ 0.16 fm−3 (Lattimer and Prakash, 2011). The



densities in CCSNe and PNSs are generally lower. Anexception is the case of so-called “failed” CCSNe leadingto BH formation (see Sec. VI.B.3). During the final collapse toa BH, densities well above 10nsat can be reached before theformation of an event horizon (Sumiyoshi, Yamada, andSuzuki, 2007; O’Connor and Ott, 2011; Hempel et al.,2012; Peres, Oertel, and Novak, 2013). However, theseextremely high densities occur only for less than a milli-second. The free-fall-like collapse to a BH is largely domi-nated by gravity and the EoS is not expected to influence itsdynamics.For the gross structure of NSs, the state of matter below

10−11 fm−3 is practically irrelevant, as it makes up only thefew outermost centimeters of the star. In CCSNe, and to someextent also in NS mergers, the situation is different. Here low-density matter plays an important role. In both cases one isinterested in the ejecta. Their densities decrease continuouslyduring their expansion. Typically, a simple ideal gas EoS isused for the description of matter under such conditions;however, full thermodynamic equilibrium (see Sec. II.B.1)cannot always be assumed. Instead a network of time-dependent nuclear reactions has to be considered. ForCCSNe, the ongoing burning processes, which contributeto the final explosion energies (Yamamoto et al., 2013; Peregoet al., 2015), occur in low-density matter. To follow theevolution of a CCSNe for several seconds, it is unavoidable toinclude a description of low-density matter out of NSE. Froma practical point of view, the connection of such a region to atabulated EoS in the higher-density and temperature regimecan be quite intricate.The temperature of a typical NS older than a few minutes is

small (below 1 MeV) on nuclear energy scales (Pons et al.,1999; Suwa, 2014) and can be considered as zero in mostapplications. However, these objects are born in CCSNewhich can be extremely hot events. The same holds forNS-NS and NS-BH mergers. Typical temperatures in CCSNeand PNS are in the range from a fraction to a few tens of MeV.This can be inferred from Fig. 1 which shows the temperaturesand densities reached during a CCSN simulation for the 15M⊙progenitor of Woosley and Weaver (1995) within the firstsecond after bounce. It is a typical example for the corecollapse of an intermediate-mass progenitor which is expectedto lead to an explosion. The temperatures obtained correspondto entropies per baryon in the range from 1 to 5 at the stage ofcollapse and up to 20 in the shock-heated matter. For otherprogenitors that are also expected to lead to explosions, therange of temperatures is similar. Scenarios with BH formationset the upper limits for density and temperature which ageneral purpose EoS has to cover. The temperature in such anevent can rise above 100 MeV; see, e.g., O’Connor and Ott(2011). Therefore the temperature domain to be covered by ageneral purpose EoS is 0 MeV≲ T ≲ 150 MeV.The color coding in Fig. 1 illustrates the electron fractions

Ye reached during the early evolution of a CCSN. The core ofthe supernova progenitor has almost an equal number ofelectrons, protons, and neutrons, i.e., Ye ≈ 0.5. During thecollapse, electron-capture reactions lead to a strong neutro-nization of matter, decreasing Ye. The presence of trappedneutrinos, which acquire a finite chemical potential, limits the

lowest electron fractions Ye which are reached in the core(Fischer et al., 2011). In its later evolution, the cooling PNSapproaches β equilibrium with neutrinos freely leaving thesystem. In the cold, final equilibrium state of a NS, the lowestelectron fractions are found to be very close to zero. In someparts of the supernova ejecta Ye can rise to values above 0.5corresponding to a proton-rich environment. Consequently,the range of a general purpose EoS to be covered is0 < Ye ≲ 0.6.The conditions in NS mergers are quite diverse. In

general they depend on the masses of the merging NSsand the EoS and also on the magnetic fields and NS spins.Typical temperatures in the core of a postmerger remnantNS are in the range from 20 to 60 MeV (Bauswein, Janka,and Oechslin, 2010). These temperatures can be wellexceeded in the contact layers in the early stage of themerger, where extremely high temperatures up to 150 MeVcan occur locally (Bauswein, Janka, and Oechslin, 2010;Rosswog, Piran, and Nakar, 2013). The highest densities inthe hot and rotating remnant NS are typically between 2nsatand 6nsat (Hotokezaka, Kiuchi et al., 2013). In case theremnant collapses to a BH, similar arguments as for failedSNe apply: during the collapse much higher densities andcorrespondingly higher temperatures are reached, but areprobably not important dynamically.The dynamic ejecta of NS mergers originate from the crust

and outer core of the merging NSs. Initially, this material hasvery low Ye in the range from 0.0 to 0.2 (Rosswog, Piran, andNakar, 2013; Sekiguchi et al., 2015). Depending on thetemperatures reached, the degeneracy of electrons is liftedand Ye increases to higher values. In the subsequent evolution,neutrino absorptions also influence Ye, resulting in finalvalues in the range of roughly 0.1 to 0.4 (Wanajo et al.,2014; Sekiguchi et al., 2015). See also Foucart et al. (2016)for a comparison of the thermodynamic conditions for differ-ent EoS. Figure 2 shows the thermodynamic conditionsreached in the remnant in the aftermath of a neutron starmerger. For the later ejecta that appear in the form of aneutrino-driven wind, extremely high entropies per baryon

FIG. 1. Temperatures and densities reached during a CCSNsimulation within 1 s post bounce. The color coding shows theelectron fraction Ye. From T. Fischer.



above 50 are found (located mostly in the polar regions),whereas most of the matter has entropies below 7, and theentropy tends to decrease with increasing density (Peregoet al., 2014; Sekiguchi et al., 2015).To conclude, we summarize in Table I the overall ranges

that have to be covered by a general purpose EoS to describeNS mergers, CCSNe, and cold NSs.

4. Particle degrees of freedom

Within these ranges of the thermodynamic variables given,the composition of matter changes dramatically. In cold NSs,heavy nuclei are present in the inner and outer crusts (Chameland Haensel, 2008). The surface layer of the outer crust ismade of 56Fe ions immersed in a sea of electrons. Withincreasing density, the nuclei become more massive andneutron rich, reaching nuclei of the neutron drip line at theboundary to the inner crust; cf. Sec. V.A.1 for details.At low densities and finite temperatures, a plasma is

expected with a mixture of nuclei, nucleons, and electrons.In the shock-heated matter of CCSNe, light nuclear clusters,such as α particles, deuterons, or tritons, are found to bethe dominant baryonic particle degrees of freedom besidesnucleons (Sumiyoshi and Röpke, 2008). At densities justbelow nuclear saturation or sufficiently high temperatures,nuclei dissolve and one is left with strongly interacting mattercomposed of nucleons and electrons. At high densities and/or

temperatures, additional particle species are expected to occur(Glendenning, 1997; Weber, 2005), such as nuclear resonan-ces, e.g., Δ baryons (Drago et al., 2014), or mesons, e.g.,pions. Also strange degrees of freedom such as hyperons(Glendenning, 1982; Chatterjee and Vidaña, 2016) or kaonscan be present. There is the possibility that the mesons formcondensates at low temperatures (Glendenning, 1997). Even atransition to deconfined quark matter (QM) is possible at highdensities and temperatures. The occurrence of antiparticles isrelevant at high temperatures, in particular, for light particlespecies. Besides electrons, muons are relevant leptonicdegrees of freedom, and so are electron-, muon-, and tau-flavor neutrinos and antineutrinos. Neutrinos are not neces-sarily in equilibrium with matter (see Sec. II.B.1). At finitetemperatures, thermal photons complete the composition.While leptons and photons can be mostly treated as freegases, this does not hold for hadrons or quarks. Theircontribution to the EoS is mainly governed by the stronginteraction deeply inside the nonperturbative regime.

III. FORMAL APPROACHES TO THE DESCRIPTION OFDENSE MATTER

The theoretical description of strongly interacting matterrequires methods which capture the essential thermodynamicproperties of the many-body system. The challenges aremultifaceted. First, the relevant degrees of freedom have tobe identified. Approaches for nuclear matter that are basedon nucleons are the predominant choice and may suffice inmany cases. It will be necessary to consider other degrees offreedom for certain thermodynamic conditions, e.g., nucleiat low temperatures and densities and hyperons or evenquarks at high temperatures and densities. Furthermore,large isospin asymmetries of the system can shift thedominating degrees of freedom to more exotic particles.Second, the interactions between the constituents have to be

FIG. 2. 2D mass histograms for (left panel) mass density ρ and electron fraction Ye or (right panel) entropy per baryon s, of a NSmerger remnant at a time of 85.4 ms after the first contact. The color coding is a measure of the amount of matter experiencing thespecific thermodynamical conditions. Adapted from Perego et al., 2014.

TABLE I. Approximate ranges of temperature, baryon numberdensity, and electron fraction a general purpose EoS has to cover tobe able to describe cold NSs, NSs in binary mergers, and CCSNe.

Quantity Range

Temperature 0 MeV ≤ T < 150 MeVBaryon number density 10−11 fm−3 < nB < 10 fm−3

Electron fraction 0 < Ye < 0.6



specified. This is a nontrivial task due to the very complexnature of the strong interaction. In addition, the representa-tion depends on the chosen degrees of freedom. In principle,one wants to describe matter directly within the well-founded theory of QCD. However, there are no ab initioQCD calculations of dense matter available at the thermo-dynamic conditions that are characteristic for compact starsor CCSNe. Even a derivation of the “true” interactionbetween nucleons or other strongly interacting hadrons fromQCD remains a very complex task despite intensive efforts.Hence, calculations have to rely on model interactions,which are partially constrained by laboratory measurements.Modern theoretical approaches aim at a derivation of inputinteractions that are systematically improvable with con-trolled uncertainties.In the next step, an appropriate method is applied to find the

actual state of the system incorporating few- and many-bodycorrelations. In the case of a phase transition, additionalthermodynamical considerations have to be applied (seeSec. III.D).The choice of the degrees of freedom, the choice of the

interaction, and the selection of the many-body method are notindependent and many different approaches exist. Here wedivide them into two categories:

(1) Ab initio many-body methods start from “realistic”few-body interactions (mainly two- and three-nucleonforces), i.e., interactions that are fitted to observablesin nucleon-nucleon scattering in vacuum and proper-ties of bound few-nucleon systems. The many-bodyproblem is then treated using different techniques, e.g.,Green’s function methods, (Dirac)-Brueckner-Hartree-Fock calculations, coupled cluster, variational, andMonte Carlo methods; see Sec. III.B.1. Some of thesemany-body methods are limited by technical prob-lems, such as the Monte Carlo methods; othersintroduce approximations, such as, for example,Brueckner-type approaches, that consider only a sub-class of all possible diagrams.

(2) Phenomenological approaches use effective inter-actions that often have a more simple structure thanrealistic interactions used in ab initio approaches.They depend on a small number of parameters,usually of the order of 10 to 15, which are fitted,in the ideal case, to different properties of severalnuclei all over the chart of nuclei and nuclear matterproperties. Typical representatives of these effectiveinteractions are the Skyrme and Gogny forces innonrelativistic calculations and meson-exchangeforces in relativistic mean-field (RMF) models; seeSec. III.B.2. Nowadays, these phenomenologicalapproaches are interpreted in terms of energy densityfunctional (EDF) theory. Applying simple many-body methods, mostly on the mean-field (MF) level,results already in a rather precise description ofnuclei and nuclear matter. The extrapolation to exoticconditions has to be considered with caution; never-theless, phenomenological approaches are the mostwidely used methods to construct EoSs for astro-physical applications.

A. Basic few-body interactions

Basic few-body interactions are the starting point ofany calculation of dense matter with ab initio many-bodymethods. The two-body interaction is largely dominant, butinteractions beyond the two-body level become important inmatter at high densities. For example, it is well known that thenuclear three-body force is essential to reproduce the satu-ration properties of nuclear matter. Forces among four or morenucleons are difficult to construct and can be neglected inmany circumstances. Realistic two- and three-body forcesbetween nucleons and hyperons, as discussed in this section,should not be confused with effective temperature anddensity-dependent interactions used in phenomenologicalmodels (see Sec. III.B.2). A more detailed survey of moderntheories for nuclear forces can be found in the review ofEpelbaum, Hammer, and Meissner (2009).Historically, Yukawa (1955) proposed the first model of the

NN interaction based on the exchange of a massive particle,the pion. His model successfully explained the range of thenuclear interaction. Since then, many phenomenologicalmodels have been developed either based on Yukawa’s ideaof meson exchange or by constructing potentials with appro-priate operator structure. With the advent of QCD as thetheory of the strong interaction in the 1970s, phenomeno-logical quark models became very fashionable, describingbaryons as quark clusters. They, however, suffer from themissing confinement and connection with QCD. Onlyrecently, with chiral effective field theories (χEFTs) and withlattice gauge theory, considerable progress has been achievedto link baryonic few-body forces to QCD.

1. Experimental data

Any theoretical model for baryonic forces can be tested bycomparing predictions to experimental data. This concernsscattering and the structure of light nuclei and hypernuclei. Inthe nuclear sector, many thousands of high-precision datapoints are available. A complete partial wave analysis ofnucleon-nucleon (NN) scattering data can be performed; seeStoks et al. (1993), Arndt, Strakovsky, and Workman (1994),Arndt et al. (2007), and Navarro Pérez, Amaro, and RuizArriola (2013) and the corresponding online databases.Deuteron properties, among others its binding energy andelectric quadrupole moment, are an important input forNN forces. Owing to the large amount of data, today’sNN interactions, phenomenological or based on effectivefield theories, have reached a very high degree of precision.The binding energies of other light nuclei and nucleon-

deuteron scattering data cannot be described satisfactorily onthe basis of a two-nucleon interaction and provide thusvaluable information on the nuclear three-body force; seeKalantar-Nayestanaki et al. (2012) for a review. Recently,properties of very neutron-rich nuclei have attracted attentionsince they provide additional constraints on the three-bodyforce; see, e.g., Wienholtz et al. (2013).For the hyperonic sector data are scarce; see Gal,

Hungerford, and Millener (2016) for a detailed review ofstrangeness in nuclear physics. Hyperon-nucleon (YN) scat-tering experiments are difficult to perform because hyperons



have very short lifetimes of the order of 10−10 s. Data onhyperon-hyperon (YY) scattering are not available. In thefitting procedure for the parameters of hyperonic two-bodyforces, in general only 35 data points from the 1960s(Engelmann et al., 1966; Alexander et al., 1968; Sechi-Zorn et al., 1968; Eisele et al., 1971) for low-energy totalcross sections in reactions involving Λ and Σ hyperons areincluded. First low-energy data on Ξ−p elastic and Ξ−p →ΛΛ scattering have been obtained at KEK (Ahn et al., 2006).In addition to scattering experiments, hypernuclear spec-

troscopy can provide valuable information. Since the firstevents recorded by Danysz and Pniewski (1953a, 1953b),many hypernuclei have been produced. These are mainlysingle-Λ hypernuclei; see, e.g., the reviews by Hashimoto andTamura (2006) and Gal, Hungerford, and Millener (2016).Some events with double-Λ hypernuclei have been detected;see Aoki et al. (1991) and Nakazawa (2010). The absence ofΣ-bound states, except for an s-wave Σ bound state in 4

ΣHe(Nagae et al., 1998), indicates a repulsive ΣN interaction (Bartet al., 1999; Saha et al., 2004; Kohno et al., 2006). Only a fewevents for Ξ hypernuclei have been observed up to now (Aokiet al., 1995; Fukuda et al., 1998; Khaustov et al., 2000).Considerable experimental efforts are underway to improvehypernuclear data; see, e.g., Agnello et al. (2012) andSugimura et al. (2014). Three-body forces are not yet wellexplored for hyperons; see, however, the recent work byLonardoni, Gandolfi, and Pederiva (2013). Hyperonic single-particle potentials in symmetric nuclear matter are often usedto determine the effective hyperon-nucleon interactions inphenomenological models; see, e.g., the discussions by Ellis,Kapusta, and Olive (1991), Glendenning and Moszkowski(1991), Schaffner et al. (1994), Balberg and Gal (1997),Glendenning (1997), Vidaña et al. (2001), and Oertel, Fantina,and Novak (2012).

2. Phenomenological forces

Approaches to obtain phenomenological forces can bedivided into two main categories: models based on mesonexchange and potential models.

a. Meson-exchange models

Meson-exchange models follow the original idea ofYukawa in which the NN interaction is mediated by mesonexchange. Additional mesons have been added to capture thecomplex dependence of the nuclear interaction on spin,isospin, and spatial coordinates. Some models have beenextended to include strange mesons in order to describe theYN and in some cases the YY interaction. The general idea isthat the pion, as the lightest particle, describes the long-rangeattractive part of the interaction and that scalar mesons areresponsible for the intermediate-range attraction whereasvector mesons govern the short-range repulsive contribution.The so-called σ meson in the scalar-isoscalar channel oftenrepresents the midrange attraction; however, its status as aparticle is very ambiguous. Many models use instead corre-lated and uncorrelated two-pion exchange to describe theintermediate range NN interaction; see, e.g., Machleidt and Li(1994) and Donoghue (2006) for discussions. The variousmodels differ mainly in the mesonic content, the treatment of

two-meson exchange, and approximations made in order toobtain practically applicable potentials from the basic ampli-tudes, for instance, the form factors used at the meson-baryoninteraction vertices. From a phenomenological point of view,the latter are introduced to account for the substructure ofbaryons. They serve as regulators in solving the scatteringequation in order to avoid any divergent contributions.Next we mention some well-known models that satisfac-

torily describe NN scattering data and the deuteron. Theclassical versions of the Nijmegen interaction for the NNsystem (Nagels, Rijken, and de Swart, 1977, 1978) are basedon the one-meson-exchange picture. They were extended toinclude YN and YY interactions (Maessen, Rijken, and deSwart, 1989; Rijken, Stoks, and Yamamoto, 1999) as well asadditional one- and two-meson exchanges (Rijken, 2006;Rijken and Yamamoto, 2006; Nagels, Rijken, andYamamoto, 2014). The Paris NN potential (Cottingham et al.,1973; Lacombe et al., 1980) uses an ad hoc parametrization atvery short distances arguing that the meson-exchange pictureis no longer valid there due to the substructure of nucleons.The Bonn potential (Machleidt, Holinde, and Elster, 1987;Machleidt, 2001) for the NN interaction is given in relativisticform in momentum space to avoid the local approximation ofnonrelativistic models, as, e.g., in the Nijmegen potentials.The Jülich group has extended the Bonn model to theYN interaction (Holzenkamp, Holinde, and Speth, 1989;Haidenbauer and Meissner, 2005).

b. Potential models

In addition to the well-established long-range one-pionexchange, potential models adopt a sum of local operators,where the essential ones are central, tensor, and spin-orbitterms. The parameters are fitted to deuteron properties andNN-scattering data. The Urbana (Lagaris and Pandharipande,1981) and Argonne potentials (Wiringa, Smith, andAinsworth, 1984; Wiringa, Stoks, and Schiavilla, 1995) areexamples of such high-quality potential models. The latestversion of the Argonne potential (Wiringa, Stoks, andSchiavilla, 1995), called v18, not only contains isoscalaroperators but includes an electromagnetic part and isovectoroperators such that the charge dependence of the NN force issuccessfully described. Also, some models with Λ hyperonsare available, but they are much less sophisticated due to thesmall amount of hyperonic data; see, e.g., Bodmer, Usmani,and Carlson (1984).Considering only two-nucleon interactions, it is well known

that light nuclei, in particular, the triton 3H, are underboundand the saturation density of nuclear matter is overestimated.This shows the importance of many-body forces to correctlydescribe nuclear systems. A major contribution to the three-nucleon force is the two-step pion exchange between twonucleons via a third nucleon that can be excited, e.g., to aΔ baryon (Fujita and Miyazawa, 1957). This feature isincorporated for instance in the Tucson-Melbourne model(Coon and Glöckle, 1981; Friar, Huber, and van Kolck, 1999;Coon and Han, 2001). Such an interaction is attractiveand helps to solve the underbinding problem in lightnuclei, whereas it worsens nuclear matter saturation proper-ties. Therefore the Urbana group proposed a series of



phenomenological three-nucleon forces, adding to the attrac-tive two-pion exchange contribution a parametrized repulsivepart (Carlson, Pandharipande, and Wiringa, 1983; Pudlineret al., 1995; Pieper et al., 2001; Pieper, 2008). The latterinteraction is adjusted to the properties of light nuclei. Theproblem of such a procedure is that the three-body force is notindependent of the two-body force employed in the fits. Morerecently, consistent two- and three-nucleon forces have beenderived within χEFT; see the next section.

3. Interactions from chiral effective field theory and lattice QCD

Since the seminal papers by Weinberg (1990, 1991) manyefforts have been devoted to the derivation of nuclear forcesfrom a χEFT. Within a chiral theory pions emerge naturally asthe relevant degrees of freedom at low energies to describe theinteraction of nucleons since they appear as Goldstone bosonsof the theory if the chiral symmetry of QCD is spontaneouslybroken. The systematic framework of effective field theoriesallows one to establish a classification of different contribu-tions to the interaction and to make the link with QCD. Thestarting point is the most general effective chiral Lagrangianthat respects the required symmetries. It is expanded in powers

of a small quantity p ∼ ðmπ=Λχ ; j~kj=ΛχÞ, where mπ denotes

the pion mass, ~k is a (soft) external momentum, and Λχ ∼1 GeV specifies the scale of chiral symmetry breaking. Inaddition to dynamical pion contributions, nucleonic contactoperators appear at each order. They contain the unresolvedshort-range physics. Their strength is controlled by so-calledlow-energy constants (LECs) that are fitted to experimentaldata. Pionless effective theories are applicable at very lowenergies; see Bedaque and van Kolck (2002) for a review.Although the details of the power counting scheme are not

yet completely settled [see, e.g., Pavón Valderrama andPhillips (2015) and references therein], chiral nuclear forceswork out well. In particular, before the advent of nuclearinteractions from χEFTs, no consistent model of three-bodyand higher many-body nuclear forces existed. In addition tothe incorporation of symmetries from QCD, the advantage ofχEFT approaches is the possibility to extend the interactionsin a consistent way to three- and many-nucleon systems.Comprehensive reviews can be found in Epelbaum, Hammer,and Meissner (2009) and Machleidt and Entem (2011). Forrecent high-quality chiral potentials, see Navarro Pérez,Amaro, and Arriola (2015) and Piarulli et al. (2015), whichincludesΔ resonances. The χEFT approach has been extendedto include strangeness and interactions of the full baryonoctet; see Polinder, Haidenbauer, and Meißner (2006) andHaidenbauer et al. (2007, and 2013). In this case, not all LECscan be determined by experiment due to the lack of relevantdata in the hyperonic sector. Instead they have partly beenfixed by flavor SUð3Þ symmetry.Another promising possibility to relate nuclear forces to

QCD is lattice QCD. In principle, it is a tool to calculatehadron properties directly from the QCD Lagrangian withMonte Carlo methods on a discretized Euclidian spacetime. Itis, however, extremely expensive in the numerical applicationeven with sophisticated state-of-the-art algorithms on highperformance computers. For the moment, simulations can be

carried out only with large quark masses and the extrapolationto physical masses is difficult. In addition, the lattice spacinghas to be fine enough and the volume large enough to avoidcomputational artifacts. Recent substantial efforts (Beaneet al., 2011; Aoki et al., 2012) give hope for future high-precision predictions. This could be interesting, in particular,for channels where only few experimental data are available,e.g., the hyperon-nucleon interaction; see, e.g., Beane et al.(2007, 2012) and Inoue et al. (2010).

4. Renormalization group methods and evolved potentials

The strongly repulsive core of two-body baryonic inter-actions renders multibaryon systems nonperturbative. Thuscorrelations become extremely important but are difficult totreat with many-body methods. The repulsive core, although adistinct feature of baryonic forces, is not directly affectinglow-energy observables. With renormalization group (RG)techniques, the high-momentum part of the interaction relatedto the repulsive core can be “integrated out” via a continuouschange in resolution by applying suitable unitary transforma-tions. In this way, the high-momentum part decouples fromthe low-momentum part and three- and many-body forcesemerge automatically from a pure two-body force. During theevolution, all generated interactions are “phase-shift equiv-alent” and low-momentum observables are preserved. Thusthe description of scattering data remains as precise as for theoriginal interaction. The obtained RG-evolved potentials aremuch more perturbative than nonevolved ones and thereforesimplify the baryonic many-body problem. In connection withmany-body perturbation theory (MBPT), i.e., a perturbativeexpansion around the Hartree-Fock (HF) solution, RG-evolved interactions became a great success for nuclearsystems; see Bogner, Furnstahl, and Schwenk (2010) andFurnstahl and Hebeler (2013) for recent reviews on thesubject.Even though all high-precision “bare” nuclear forces are

rather different, almost unique RG-evolved potentials emergeat low momenta (Schwenk, 2005), often denoted as V low-k.Schaefer et al. (2006) and Wagner et al. (2006) applied thesame techniques to nucleon-hyperon interactions. It turned outthat the resulting low-momentum interactions are differentfrom each other because the bare potentials are much lessconstrained. Hence, it is not surprising that there is a largespread in the results if they are applied to dense matter withhyperons (Đapo, Schaefer, and Wambach, 2008, and 2010).This shows the lack of relevant experimental data concerningthe hyperon-nucleon and hyperon-hyperon interactions.

B. Many-body methods for homogeneous matter

The first step in studying strongly interacting matter is oftenthe investigation of homogeneous matter at vanishing temper-ature where almost all methods discussed later can be applied.Even if the basic few-body interactions were exactly known,the theoretical modeling is not a trivial task since any naiveperturbative expansion is likely to fail.The most simple method to treat the many-body problem

beyond the perturbative level is the HF approximation; seeSec. III.B.2 and Fetter and Walecka (1971), Ring and Schuck



(1980), and Greiner and Maruhn (1996) for more details. Theidea is that each particle moves in a single-particle potential,the “mean field,” generated by the average interaction withall other particles. In practice, the many-body wave functionis approximated as an antisymmetrized product of single-particle wave functions, which are determined self-consistently. Although generally successful in atomic physicsand in chemistry, in a nuclear system a HF calculation startingfrom conventional two-body interactions fails to reproduceknown properties of nuclear matter. In addition, the results aresensitive to the modeling of the short-range repulsive core ofthe two-body interaction which is not fixed uniquely byscattering data. This can be understood since the HF approxi-mation neglects any short-range correlations between theparticles which arise from their mutual interaction.Two ways out of this problem are currently applied: either

correlations are explicitly included within the many-bodyapproach or, instead of realistic few-body interactions, aneffective, usually medium dependent, interaction is usedwithin a HF approach.In Sec. III.B.1, we discuss different theoretical ab initio

frameworks to include correlations in a strongly interactingmany-body system. Interesting attempts to compare in aquantitative way different ab initio many-body methods canbe found, e.g., in Baldo and Maieron (2004), Bombaci et al.(2005), and Baldo et al. (2012). Section III.B.2 is devoted tomodels with different types of phenomenological effectiveinteractions. Apart from the textbooks, there exist manyexcellent reviews on the different standard many-body meth-ods; see, e.g., Müther and Polls (2000), Baldo and Burgio(2012), and Carlson et al. (2015). Therefore we do not aim togive a comprehensive and complete overview, but present onlythe general ideas.

1. Ab initio methods

a. Self-consistent Green’s function

The idea of the self-consistent Green’s function (SCGF)method is that the system’s energy can be calculated con-veniently from the single-particle Green’s function G. Itdescribes the propagation of a single-particle state ψ fromtime t and position ~x to t0 and ~x0 as

ψðt0; ~x0Þ ¼Z

d4xGðt0; ~x0; t; ~xÞψðt; ~xÞ: ð3Þ

The level of approximation in the SCGF method is controlledby the approximations made in order to determine the single-particle Green’s function. A thorough definition can be foundin any textbook on quantum theory at finite density andtemperature; see, e.g., Fetter and Walecka (1971). For anoninteracting homogeneous system at zero temperature theGreen’s function can be written in momentum space as2

G0ð~k;ωÞ ¼ θðj~kj − kFÞω − E0ð~kÞ þ iη

þ θðkF − j~kjÞω − E0ð~kÞ − iη

; ð4Þ

where kF denotes the Fermi momentum and E0ð~kÞ ¼ ~k2=ð2mÞdenotes the noninteracting single-particle energy of a particlewith mass m. Any indices related to further quantum numbersof the particle are suppressed for clarity. The first term on theright-hand side (rhs) of Eq. (4) describes the propagation of astate outside the Fermi sea, a particle, and the second term astate inside the Fermi sea. Since per definition all states in theFermi sea are filled, it can propagate inside the Fermi sea onlyas a hole, i.e., a particle removed from the Fermi sea.The energy density of the system can be straightforwardly

calculated from the trace of the single-particle Green’sfunction. For the noninteracting system the well-knownexpression for an ideal Fermi gas is obtained. Of course, adense baryonic system cannot be described as a Fermi gas ofnoninteracting particles. Thus the full interacting single-particle Green’s function G has to be determined. A calcu-lation from a perturbative series in the interaction potential isnot viable in view of the strong baryonic interaction. At thispoint self-consistency is introduced in the form of Dyson’sequation, which schematically can be written as

G ¼ G0 þ G0ΣG ð5Þ

with the one-particle irreducible3 self-energy Σ which itself isdetermined by the interaction.Retaining the lowest-order diagrams in the interaction, the

HF approximation can be derived with the SCGF formalism.Formally this means that in the HF approximation theN-particle Green’s functions are (antisymmetrized) productsof single-particle Green’s functions. When going beyond theHF approximation, the dominant effect should be multiplescattering processes with two participating baryons. Underthis assumption the so-called “ladder approximation” isobtained. The name originates from a diagrammatic repre-sentation of this approximation for the single-particle Green’sfunction depicted in Fig. 3. In practice, the complete two-particle T matrix is introduced, describing an effective two-particle interaction upon summing up all “ladders”; see Fig. 3.Note that the ladder approximation includes the two contri-butions leading to the HF approximation. The equation for theT matrix can be written as

h12jTj34i ¼ h12jVj34i þXnn0

h12jVjnn0iGnGn0 hnn0jTj34i;

ð6Þ

where V represents the bare two-body interaction and asummation over intermediate states has to be performed.In the intermediate states of the T matrix there can be no

propagation of a particle and a hole state. In the ladder2For simplicity we assume nonrelativistic kinematics. A relativ-

istic treatment does not change the general reasoning. We furtherconsider the zero temperature limit. Finite temperature is easilyincluded in the formalism; see, e.g., Fetter and Walecka (1971) andFrick (2004).

3One-particle irreducible means that the diagrams cannot bedisconnected by cutting a fermion line. It is obvious that the self-consistent resummation via Dyson’s equation automatically gener-ates all reducible terms.



approximation the self-consistent Green’s function methodthus sums up particle-particle and hole-hole ladders to allorders. Physically the ladder diagrams take into accountmultiple scattering of particles and holes and can therefore,in contrast to the HF approximation, describe the effect thatthe strong short-range repulsion disfavors states when twoparticles come very close to each other. Reviews on theapplications of the method to nuclear problems can be foundin Müther and Polls (2000) and Dickhoff and Barbieri (2004).The results for the EoS improve substantially in comparison tothe HF approximation.The choice of the contributions included in the summation

(ladder, ring, parquet) has been more or less intuitive and wasjustified by the results. It is not clear how the method can besystematically extended in order to improve the resultsachieving self-consistency and avoiding double counting.So far three-nucleon forces were not included in the ladderapproximation although their importance in nuclear systemsis well known. A method to include them in the SCGFformalism was developed recently by Carbone et al. (2013).Another point is that low-temperature nuclear (and bar-

yonic) matter is unstable with respect to the formation of asuperfluid or superconducting state. This is the well-knownCooper instability (Cooper, 1956): a fermionic many-bodysystem with an attractive interaction tends to form pairs at theFermi surface. The instability shows up as a pole in theT matrix when the temperature falls below the criticaltemperature Tc for the transition to the superfluid or super-conducting state; see, e.g., Thouless (1960), Schmidt, Röpke,and Schulz (1990), Alm et al. (1993), and Stein et al. (1995).Formally, the approach can be extended to include thepossibility of superfluidity and superconductivity by intro-ducing anomalous Green’s functions describing pair forma-tion. However, practical SCGF calculations in the ladderapproximation are numerically already very demanding sincethe full energy dependence of the intermediate states has to beaccounted for. Therefore actual calculations are often per-formed at temperatures above Tc and extrapolated to zerotemperature; see, e.g., Frick (2004).

b. Brueckner-Hartree-Fock

The Brueckner-Hartree-Fock (BHF) approximation is awidely used microscopic many-body method developed byBrueckner, Bethe, and others in the 1950s. Numerically it isless involved than the SCGF discussed previously. In a generalframework it can be derived from the Brueckner-Bethe-Goldstone hole-line expansion, truncated at the two-hole-line

level for the evaluation of the ground-state energy. A “holeline” has to be considered as the propagation of a hole. Thisseries can be roughly understood as an expansion in density(Fetter and Walecka, 1971). At low densities the contributionswith an increasing number of hole lines should be suppressed,thus ensuring good convergence. It is generally assumed thatthe n-hole-line contributions contain the dominant part of then-body correlations. Detailed and pedagogical introductionscan be found in Day (1967), Fetter and Walecka (1971), andBaldo and Burgio (2001, 2012).The BHF method can be obtained from the SCGF approach

in the ladder approximation after some simplifications. Thefirst one is to neglect the hole-hole contributions. The secondone is to approximate the full self-energy of the intermediatestates by a quasiparticle approximation. The equation for theT matrix, Eq. (6), then becomes an equation for the BruecknerGmatrix by replacing T with G, except that the product of thetwo single-particle Green’s functions in the intermediate statesis approximated by

GnGn0 →Pðn; n0Þ

ω − En − En0 þ iη; ð7Þ

where the Pauli operator P is nonzero only if both states lieoutside the Fermi sea, i.e., they correspond to two particlestates. Note that the denominator does not contain the full self-energy. In the quasiparticle approximation it has the form ofthe noninteracting system.The single-particle energies E are determined self-

consistently from the G matrix in the following way (writtenin momentum space):

Eð~kÞ ¼~k2

2mþ Uð~kÞ; ð8Þ

Uð~kÞ ¼X

j ~k0 j<kF

hkk0jG(Eð~kÞ þ Eð ~k0Þ)jkk0iA; ð9Þ

where the subscript A indicates that the matrix element has tobe antisymmetrized.The BHF method is not fully self-consistent. There remains

some freedom in determining the single-particle energy Eð~kÞ;see Baldo and Burgio (2001) for a discussion. It was shown byBaldo et al. (2001) that with a proper choice the three-hole linecorrections to the energy are small, indicating good conver-gence of the series. The main problem is that the BHF methodviolates the Hugenholtz–van Hove theorem (Hugenholtz andvan Hove, 1958) and, hence, is thermodynamically incon-sistent. This theorem states that the single-particle energy at theFermi surface should equal the chemical potential. Numericallythe differences are of the order of 10–20 MeV at saturationdensity (Bożek and Czerski, 2001).In the BHF approximation, the description of nuclear matter

is improved substantially as compared with the HF calcu-lations. However, the saturation properties are not satisfac-torily reproduced with only two-body forces. It is generallybelieved that three-body forces are needed. In nonrelativisticBHF calculations they can be explicitly included; see, e.g.,Lejeune, Lombardo, and Zuo (2000) and Zuo et al. (2002a,

FIG. 3. Feynman diagrams illustrating the lowest-order contri-butions to the self-energy in Dyson’s equation (5) for the single-particle Green’s function in the ladder approximation (withoutexchange contributions). Solid lines represent fermion propaga-tors and wavy lines an interaction.



2002b). So far, there is no general consensus on how toimprove BHF calculations systematically such that uncertain-ties are under control.It is possible to extend the nonrelativistic BHF formalism in

order to treat baryons in a special relativistic way. This Dirac-Brueckner-Hartree-Fock (DBHF) approach (Brockmann andMachleidt, 1984; Horowitz and Serot, 1987; Ter Haar andMalfliet, 1987; Brockmann and Machleidt, 1990;Sammarruca, 2010) is computationally more involved thannonrelativistic BHF calculations and some ambiguities existconcerning the representation of the in-medium G matrix interms of Lorentz invariants; see, e.g., the discussion by Gross-Boelting, Fuchs, and Faessler (1999). However, the mainadvantage is that an additional repulsion at high densities isobtained since part of the three-body interaction is automati-cally generated (Brown, 1987). Relativistic DBHF approachesalso avoid the problem of nonrelativistic BHF calculationswhich can result in a superluminal speed of sound at the highcentral densities of massive NSs.

c. Methods derived from the variational principle

The Ritz-Raleigh variational principle is the basis forvariational approaches to the many-body problem. It ensuresthat the trial ground-state energy

Etrial ¼hΨtrialjHjΨtrialihΨtrialjΨtriali

; ð10Þ

calculated from the system’s HamiltonianH with a trial many-body wave function Ψtrial, gives an upper bound for the trueground-state energy of the system. Correlations can beembodied in the trial wave function, given for the A-baryonsystem by

Ψtrialð1;…; AÞ ¼ Fð1;…; AÞΦMFð1;…; AÞ: ð11Þ

The operator F is intended to transform the uncorrelated wavefunction ΦMFð1;…; AÞ to the correlated one. ΦMF is anantisymmetrized product (a Slater determinant) of single-particle wave functions. In practice, an ansatz is chosen for thetrial wave function and its parameters are varied in order tominimize Etrial. Once the trial wave function is determined,expectation values of other operators can be evaluated.The idea of the variational method is very simple and

appealing. The difficulty resides, however, in the details,namely, the numerical evaluation of the different expectationvalues. The first point is the choice of the interactionHamiltonian. As it stands, the method is conceived for treatinga local nonrelativistic potential.4 Thus, some of the realisticpotentials discussed in Sec. III.A, in particular, those involv-ing energy-dependent meson exchange, cannot be used withinthis approach. There are, however, potentials which have beendesigned specifically for variational methods. The mostprominent example is the series of Argonne NN forces (seeSec. III.A.2.b). Most variational calculations include a three-body force in the nuclear Hamiltonian. Together with the

Argonne forces, the Urbana three-body forces are usuallyapplied. Further, Gezerlis et al. (2013, 2014) developed a localversion of nuclear interactions for quantum Monte Carlocalculations from χEFT which in principle is applicable.The central task is to find a suitable ansatz for the

correlation operator F. At sufficiently low densities, two-body correlations should be dominant in nuclear systems. Thisassumption represents, for instance, the basis for the ladderapproximation discussed in Sec. III.B.1.a. Within the varia-tional methods this assumption leads to an ansatz for thetwo-body correlation operator F2 which contains essentiallythe same operational structure as the two-body interaction(Fantoni and Fabrocini, 1998; Müther and Polls, 2000;Carlson et al., 2015). Thus, it is written as a sum of two-body operators, incorporating the nuclear spin-isospindependence, multiplied by radial correlation functionsfðmÞðrÞ:

F2ði; jÞ ¼Xm

fðmÞðrijÞOðmÞði; jÞ: ð12Þ

The fðmÞðrÞ are then determined by minimizing Etrial. Inpractice, this is done employing different techniques.In the nuclear context, Fermi-hypernetted-chain (Fantoni

and Rosati, 1975; Pandharipande and Wiringa, 1979) calcu-lations have proven to be efficient. We note that up to now it isimpossible to include the complete operator structure of themost sophisticated Argonne potentials in the correlationoperators F. The spin-orbit correlation, in particular, cannotbe treated on the same footing, since it cannot be chained.In coupled cluster theory, which is based on ideas of

Coester (1958) and Coester and Kümmel (1960), the corre-lation operator F is represented in an exponential form F ¼expðTÞ with the cluster operator

T ¼XAm¼1

Tm ð13Þ

that is a sum ofm-particlem-hole excitation operators Tm. Themethod, which is nonvariational in practice, is utilized withgreat success in quantum chemistry (Bartlett and Musial,2007) and nuclear structure calculations (Hagen et al., 2007,2010, 2012). Early applications to nuclear matter can be foundin Kümmel, Lührmann, and Zabolitzky (1978) and Day andZabolitzky (1981). Nowadays, optimized chiral nucleon-nucleon interactions are implemented (Baardsen et al.,2013; Hagen et al., 2014).The variational ground-state energy represents only an

upper bound on the exact ground-state energy, and thedeviation depends on the choice of the trial wave function.The method of correlated basis functions (CBF) allows one toimprove on the variational ground state (Fantoni andFabrocini, 1998). The idea is to add up second-order pertur-bative corrections to the ground-state energy calculated withcorrelated basis functions. The latter are determined by thevariational calculation from model basis functions. Takingonly ΦMF as a model basis function, the usual variationalcalculation would be recovered. Within CBF, other basis

4SeeWalhout et al. (1996) for an attempt to generalize this methodto relativistic systems within the path integral formalism.



functions are added, containing already some particle-holeexcitations on the initial wave function ΦMF.Another widely used method in nuclear physics to evaluate

expectation values is the variational Monte Carlo approach;see the reviews by Guardiola (1998) and Carlson et al. (2015).Sophisticated calculations with this technique have beenperformed for light nuclei, including two-body correlations,such as given in Eq. (12), and triplet correlations; see, e.g.,Wiringa et al. (2014). Since the computational effort increasesvery rapidly with the number of nucleons, the EoS ofhomogeneous nuclear matter, however, is extremely difficultto obtain.

d. Quantum Monte Carlo methods

Advancing computer technology has allowed for rapidprogress in the application of Monte Carlo methods to nuclearsystems in recent years. In addition to the variationalMonte Carlo approach discussed in the previous section,the ground-state wave function and energy can be determinedby evolving the many-body Schrödinger equation in imagi-nary time. Monte Carlo sampling is thereby used to evaluatethe paths. In nuclear physics these methods suffer, however,from the fermion sign problem and different approximationsare employed; for comprehensive reviews, see Guardiola(1998), Carlson, Gandolfi, and Gezerlis (2012), andCarlson et al. (2015).The Green’s function Monte Carlo (GFMC) method

(Carlson, 1987, 1988) gives accurate results for light nuclei,but due to the nuclear spin and isospin degrees of freedom,computing time increases exponentially with the number ofparticles. Up to now, the largest systems treated are 12C and asystem of 16 neutrons. The auxiliary field quantumMonte Carlo (AFQMC) approach (Schmidt and Fantoni,1999) introduces auxiliary fields by Hubbard-Stratonovichtransformations to sample the spin-isospin states. This effi-cient sampling allows for treating larger systems, with morethan 100 nucleons. Finite-size effects are expected to be smalland AFQMC calculations have been applied in the last yearsto homogeneous matter, in particular, neutron matter, but alsoto symmetric matter and nuclear matter with hyperons (seeSec. IV.B).In spite of the recent progress, it is still not possible to

perform GFMC and AFQMC calculations with the fullArgonne v18 potential since some of the terms, again relatedto the spin-orbit structure, induce very large statistical errors.Simplified potentials have been developed (Pudliner et al.,1997; Wiringa and Pieper, 2002) containing less operatorswith readjusted parameters. Besides these two-body inter-actions, the Urbana three-body potentials are used. Recently, alocal chiral potential was developed (Gezerlis et al., 2013)which is well suited for quantum Monte Carlo techniques.

e. Chiral effective field theory

χEFT is very successful in describing nuclear forces andhas allowed one to establish a link between the underlyingtheory of QCD and nuclear physics (see Sec. III.A.3). Exceptfor very light nuclei, where direct numerical solutions of theSchrödinger equation are possible, these chiral forces aregenerally employed within standard many-body techniques to

address heavier nuclei or homogeneous nuclear matter. Inrecent years some effort has been devoted to an alternativeapproach for homogeneous matter, namely, extending theidea of chiral perturbation theory directly to nuclear mattercalculations, i.e., developing an effective field theory (EFT)for nuclear matter. Similar to nuclear forces in vacuum, pionsand nucleons are treated as explicit degrees of freedom andshort-range dynamics is comprised in local contact terms. Theadvantage of such an EFT is that it establishes a powercounting which allows one to select at a given order therelevant ones among the infinite number of contributions andthat one can determine an associated uncertainty, which is notpossible in many other methods. The main difficulty resides indefining a well-adapted power counting scheme.For nuclear matter, the nuclear Fermi momentum kF enters

as an additional scale. It is considered as small, of the sameorder as the pion mass. At saturation density it is given askF ≈ 263 MeV, which is indeed smaller than a typical had-ronic scale. Based on this assumption, different power countingschemes have been developed.On the one hand, in theworks ofKaiser, Fritsch, and Weise (2002, 2003, 2005), the vacuumchiral power counting has been applied directly to nuclearmatter. On the other hand, Meißner, Oller, and Wirzba (2002),Oller, Lacour, and Meißner (2010), and Lacour, Oller, andMeißner (2011) argue that a propagating nucleon in themedium cannot always be counted in the standard way as

1=j~kjwith j~kj being a typical nucleon three-momentum, but thatthere are “nonstandard” situations where it is to be counted asan inverse nucleon kinetic energy. In practice, within thisnonstandard counting, certain classes of two-nucleon diagramshave to be resummed. Another difference is that these worksinclude local nucleon-nucleon interactions fixed by freenucleon-nucleon scattering in addition to nucleon interactionsmediated by pion exchange (Oller, Lacour, andMeißner, 2010;Lacour, Oller, and Meißner, 2011).

f. Lattice methods

The ab initio approach to solve QCD numerically becomesextremely complicated at finite densities. The fermion deter-minant in the medium turns complex valued due to theappearance of the chemical potential. Consequently theintegrals, which are evaluated in lattice QCD withMonte Carlo methods, have no longer positive weights.Different approaches to avoid this fermion sign problem havebeen suggested, e.g., reweighting techniques (Fodor and Katz,2002), the introduction of complex chemical potentials(de Forcrand and Philipsen, 2002), or Taylor expansionschemes (Allton et al., 2002). A crucial parameter in all ofthese approaches is μB=T, which for these analyses is typicallyof magnitude 1 or less. However, this value is significantlylarger in the applications we are interested in, in particular, forNSs. Accordingly, no consistent cold matter or supernova EoShas been provided by lattice QCD so far.In recent years lattice methods have been applied directly to

nuclear systems; see Lee (2009) for a review. In the so-called“nuclear lattice effective field theory” (NLEFT), nucleons aretreated as pointlike particles residing on the lattice sites. Forthe interactions, EFT nuclear forces are employed consistingof nucleon contact terms and potentially pion exchanges.



These are represented on the lattice as insertions on thenucleon world lines. Because of the approximate SUð4Þ spin-isospin symmetry of nuclear forces, NLEFT suffers much lessfrom the sign problem than lattice QCD. These methods havebeen applied successfully to light and medium mass nuclei(Epelbaum et al., 2014; Lähde et al., 2014), and to diluteneutron matter up to roughly one-tenth of nuclear mattersaturation density (Lee, 2009). For the moment, however,there are no computations of denser systems. Lattice methodsare used in the context of quantum Monte Carlo simulations,too (Wlazłowski et al., 2014).

g. Perturbative QCD

QCD is asymptotically free, viz., the coupling constantdecreases logarithmically with the energy (Gross andWilczek,1973; Politzer, 1974) and therefore, it is addressable byperturbative methods if the coupling constant turns smallenough. This has been exploited to describe the thermody-namics of dense deconfined quark matter at finite temperature(Freedman and McLerran, 1977a, 1977b, 1977c). Recentefforts aimed to account for all second-order effects in anexpansion of the thermodynamic pressure of deconfined QCD(Kurkela, Romatschke, and Vuorinen, 2010). This proceduregives valuable insights into the high-density limit of QCD andtherefore provides an important constraint for the asymptoticbehavior of the EoS at baryonic chemical potentials of severalGeV (Kurkela et al., 2014). However, results from matchingthe EoS obtained within this approach directly to a nuclearmatter EoS as suggested by Kurkela, Romatschke, andVuorinen (2010) do not provide a conclusive answer as inthis domain nonperturbative features cannot be neglected.

h. Dyson-Schwinger approach

The Dyson-Schwinger (DS) formalism is a nonperturbativeapproach to analyze QCD. It starts from a generating func-tional (the partition function of QCD). From there, coupledintegral equations, the DS equations, are derived for the n-point Schwinger functions of the theory. Formally, thisapproach is similar to that of self-consistent Green’s functionswe discussed in Sec. III.B.1.a, but now based on the QCDLagrangian. A further successful proving ground for thisapproach is QED. As in any many-body theory, every QCDSchwinger function couples to further Schwinger functions ofhigher order. This implies an infinite hierarchy of DSequations which can be solved practically only by introducingtruncation schemes. There is no strict prescription how totruncate without erasing inherent properties (e.g., symmetries)of the original theory. The truncation scheme defines a specificmodel which then can be compared to experimental data andsubsequently used to predict observables. The theoreticalframework for vacuum and in-medium studies and numerousapplications have been reviewed in detail in Roberts andSchmidt (2000), Alkofer and von Smekal (2001), and Roberts(2012). Despite the number of successful vacuum studies atzero and finite temperature it has been used only ratherrecently to compute EoSs of dense homogeneous quark matterin the deconfined phase (Chen et al., 2008, 2011, 2015; Klähnet al., 2010). Prominent topics are superconducting phases;see, e.g., Nickel, Wambach, and Alkofer (2006) and Alford

et al. (2008), and the role of strange quarks (Nickel, Alkofer,and Wambach, 2006; Müller, Buballa, and Wambach, 2013).Further DS studies investigate the critical line in the QCDphase diagram; see, e.g., Fischer, Luecker, and Mueller(2011), Qin et al. (2011), Bashir et al. (2012), andGutierrez et al. (2014). Although the DS approach promisesinsights from a QCD-based framework, no EoS has beenobtained to date that covers the whole parameter spacerequired to perform CCSN simulations. However, Klähnand Fischer (2015) showed that both the Nambu–Jona-Lasinio (NJL) model and the thermodynamic bag model(see Sec. III.B.2.b) can be understood as solutions of in-medium DS equations in rainbow approximation assuming acontact interaction for the gluon propagator.

2. Phenomenological approaches

Phenomenological approaches to describe dense matter arecharacterized by the use of effective interparticle interactionsinstead of realistic forces. They usually have a rather simplefunctional form in order to allow them to be used in severalapplications, i.e., not only uniform matter but in many casesalso finite nuclei. However, their structure can be guided bysymmetry principles, power counting arguments or insightsfrom ab initio approaches. Effective Hamiltonians can bederived from more fundamental forces using the formalism ofdensity-matrix expansions (Negele and Vautherin, 1972;Dobaczewski, Carlsson, and Kortelainen, 2010; Stoitsov et al.,2010). In general, the actual parameters of the interactions arenot directly calculated from underlying fundamental theoriesbut they are determined empirically by fitting to (pseudo-)observables that are calculated in certain approximations ofmany-body theory. The interaction and the considered modelspace are not independent. Systematic extensions of theempirical approaches are not straightforward and usuallyrequire a refit of the model parameters.Most empirical descriptions of dense matter rely on the MF

approximation or use the closely related language of energydensity functionals. Originally, the mean-field approximationcorresponded to the Hartree approximation of the many-bodystate, i.e., a simple product of single-particle wave functions,but the term is often used to denote the HF approximation, too.The constituents are considered as quasiparticles with modi-fied properties, e.g., effective masses that are different fromtheir rest masses in the vacuum. Quantities such as the energydensity or the pressure of the system can be expressed asfunctionals of the single-particle densities and an EDF isderived. It can be used as a starting point for the developmentof more refined EDFs that take into account features such asexchange and correlation effects going beyond the mean-fieldapproximation. It is well known from the basic theorems indensity functional theory (Hohenberg and Kohn, 1964; Kohnand Sham, 1965; Dreizler and Gross, 1990; Kohn, 1999;Fiolhais, Nogueira, and Marques, 2003) that an EDF existswhich yields the exact energy of the system’s ground state butits explicit form is not known. With suitable extensions ofmean-field EDFs guided by empirical information one can tryto come close to the exact EDF of the system, even if theinteraction itself is not completely known.



A further set of phenomenological EoSs can be charac-terized as purely parametric models which by themselves arenot based on a description of the interaction of particles.Instead, a parametrized functional, typically of the energydensity, is either assumed or fitted to microscopically moti-vated EoSs. These models are useful to analyze astrophysicaldata. Examples are single and piecewise polytrope fits fornuclear matter (Read et al., 2009) and a linear fit for quarkmatter (Zdunik, 2000; Alford, Han, and Prakash, 2013).

a. Hadronic matter

Self-consistent MF models are well developed. They aresuccessful in describing the properties of systems composedof nucleons and were first used for bulk nuclear matter.Nowadays they are mainly applied in the description of finitenuclei; see, e.g., the review by Bender, Heenen, and Reinhard(2003). The approaches can be divided into two main classes:nonrelativistic and relativistic models. The main distinctionsbetween them are the specific form of the interaction and theresulting dispersion relation of the quasiparticles.Nonrelativistic approaches generally start from a

Hamiltonian

H ¼ T þ V ð14Þ

for the many-body system that contains the usual kineticcontribution

T ¼Xi

p2i

2mið15Þ

and a potential term V that varies from model to model.Usually it is given as a sum

V ¼Xi<j

Vij þXi<j<k

Vijk ð16Þ

of two-body (Vij) and three-body (Vijk) interactions. Thelatter are required in order to reproduce the empiricalsaturation properties of nuclear matter. The energy of thesystem is calculated under certain assumptions for the form ofthe many-body wave function, usually within the HF approxi-mation. Pairing effects can be considered in the Hartree-Fock–Bogoliubov approximation. However, the original modelinteraction V cannot always be used in the pairing channeland a suitable pairing interaction has to be specified sepa-rately. Nonrelativistic approaches are in danger failing in thedescription of dense matter at high densities, e.g., the EoS canbecome superluminal.Relativistic models are commonly formulated in a field-

theoretical language by defining a Lagrangian density L thatserves as the starting point in order to derive the fieldequations of the interacting particles. They constitute a setof coupled equations that have to be solved self-consistently.Expressions for the energy density and pressure are obtainedfrom the energy-momentum tensor.The foremost application of MF models is the description of

finite atomic nuclei but nuclear matter properties are easily

obtained once the parameters of the effective model inter-action are determined. Depending on the selection of observ-ables and preferences in the fitting of the effective interaction,different parametrizations are obtained. For the most commonMF approaches, several hundred parameter sets are availablein the literature. In the following, the most-used MF models,distinguished by the choice of the interaction, are considered.

• Mean-field models with Skyrme-type interactions: Aneffective zero-range interaction for HF calculations wasintroduced by Skyrme (1956, 1959). After the pioneeringcalculations of nuclei by Vautherin and Brink (1970,1972), Brack and Quentin (1974b), and Beiner et al.(1975), it became very popular and found widespreaduse; see, e.g., the review article by Stone and Reinhard(2007). The basic form of the Skyrme interactionbetween nucleons 1 and 2 can be written as

VðSkyrmeÞ12 ¼ t0ð1þ x0PσÞδðr12Þ

þ 12ð1þ x1PσÞ½ðk†Þ2δðr12Þ þ δðr12Þk2�

þ t2ð1þ x2PσÞk† · δðr12Þkþ 1

6t3ð1þ x3PσÞδðr12ÞnαBðR12Þ

þ iW0ðσ1 þ σ2Þ · k†δðr12Þk ð17Þ

with parameters ti, xi, α, and W0. In Eq. (17) Pσ ¼ð1þ σ1 · σ2Þ=2 denotes the spin-exchange operator, k ¼ð∇1 −∇2Þ=ð2iÞ is the relative momentum, and nB is thetotal nucleon density. The relative coordinate and center-of-mass coordinate are defined by r12 ¼ r1 − r2 andR12 ¼ ðr1 þ r2Þ=2, respectively. The contribution withfactor t3 is a generalization that originates from anexplicit three-body term

V123 ¼ t3δðr1 − r2Þδðr2 − r3Þ ð18Þ

in the original Skyrme interaction and was converted to adensity-dependent two-body interaction. The parameterα controls the strength of the repulsion. The originalthree-body interaction (18) corresponds to α ¼ 2. Thecontribution with the factor W0 generates the spin-orbitinteraction in systems that are not spin saturated. For thedescription of nuclei, contributions from the Coulombinteraction have to be considered in addition. Thepotential (17) can be seen as an expansion in powersof the relative momentum k. Since it stops at secondorder, the interaction cannot be applied reliably in caseswhere the momenta of the nucleons reach high values,e.g., in nuclear matter at densities substantially abovesaturation.

Evaluating the energy density from the Hamiltonian inHF approximation yields an EDF that depends on anumber of single-particle densities and their spatialderivatives. Since the interaction is of zero range,exchange contributions are easily obtained and onlylocal densities appear in the EDF. Besides the usualsingle-particle number densities ni, the kinetic-energydensities τi, the currents ji, the spin-orbit densities Ji, thespin densities σi, etc. are relevant; see, e.g., Bender,



Heenen, and Reinhard (2003) for details. In applicationsto nuclear matter, currents, spin densities, and spatialderivatives of all single-particle densities vanish. Theenergy density becomes a more-or-less simple functionalin fractional powers of the number densities (Dutra et al.,2012). Obviously, an extrapolation to high densities canlead to divergences. Several extensions of the standardSkyrme functional have been proposed; see, e.g.,Lesinski et al. (2007), Bender et al. (2009), Margueronet al. (2009, 2010, 2012), Margueron and Sagawa(2009), Chamel et al. (2011), Dutra et al. (2012),Hellemans, Heenen, and Bender (2012), and Davesneet al. (2015). We note that not every Skyrme-type EDFcan be derived from simple two- and three-body poten-tials in a MF approximation. A number of well-calibratedparametrizations were proposed recently (Chabanat et al.,1997, 1998; Agrawal, Shlomo, and Au, 2005; Stone,2005; Goriely, Chamel, and Pearson, 2009a, 2009b,2013a, 2013b; Kortelainen et al., 2010, 2012, 2014;Washiyama et al., 2012). They predict many propertiesof nuclei close to experimental values with rather smalldeviations. The performance of 240 Skyrme parametri-zations under nuclear matter constraints was studied byRikovska Stone et al. (2003) and Dutra et al. (2012).Only a few satisfy all criteria that were selected. SomeSkyrme forces show instabilities (Chamel and Goriely,2010; Kortelainen and Lesinski, 2010; Hellemans et al.,2013; Navarro and Polls, 2013; Pastore et al., 2014)under particular conditions that may be cured withappropriate modifications of the EDF.

• Mean-field models with Gogny interaction: Instead of azero-range force as in the Skyrme case, the use of finite-range interactions is an established approach in MFmodels. A sum of two Gaussians was suggested by Brinkand Boeker (1967) for HF calculations. A review ofphenomenological interactions in early HF models wasgiven by Quentin and Flocard (1978). In order to obtainquantitatively reasonable results, a density-dependenteffective two-body interaction was added by Gognywhich leads to the present form

VðGognyÞ12 ¼

Xj¼1;2

exp

�−r212μ2j

�

× ðWj þ BjPσ −HjPτ −MjPσPτÞþ t3ð1þ x0PσÞδðr12ÞnαBðR12Þþ iWlsðσ1 þ σ2Þ · k†δðr12Þk ð19Þ

with parameters μj, Wj, Bj, Hj, Mj, t3, α, and Wls, andthe isospin-exchange operator Pτ ¼ ð1þ τ1 · τ2Þ=2(Dechargé and Gogny, 1980). The density-dependentand the spin-orbit contributions have the form of thecorresponding terms in the Skyrme interaction, althoughwith a different notation of the parameters. Because ofthe finite-range part in the Gogny interaction, it istechnically more involved to consider the exchangecontributions to the energy density. On the other hand,divergences of a zero-range interaction are avoided incalculations involving pairing channels. Because of the

more involved numerical calculations, there are only fewparametrizations of the Gogny interaction that are used inpractice (Goriely, Hilaire, and Koning, 2008). A collec-tion of these parameter sets with a comparison topredictions of nuclear matter properties can be foundin Sellahewa and Rios (2014).

• Relativistic mean-field and Hartree-Fock models: Inrelativistic approaches to nuclear matter and finite nuclei,a field-theoretical formalism is employed where nucle-ons are represented by Dirac four spinors ψ i and thenucleon-nucleon interaction is modeled by an exchangeof mesons. This description, called quantum hadrody-namics (QHD), was originally seen as a fully field-theoretical approach (Fetter and Walecka, 1971; Chinand Walecka, 1974; Walecka, 1974; Serot and Walecka,1986; Serot, 1992) and treated with the respectiveformalism. Later, the view of an effective descriptionto be applied in rather simple approximations prevailedsince nucleons as composite objects cannot be consid-ered as fundamental degrees of freedom. The commonstarting point in QHD models is a Lagrangian

L ¼ Lnuc þ Lmes þ Lint ð20Þ

that contains contributions of nucleons i

Lnuc ¼Xi¼n;p

ψ iðγμi∂μ −miÞψ i ð21Þ

with rest mass mi, of free mesons Lmes, and an inter-action term Lint.

In early versions of the model only isoscalar mesonssuch as the (Lorentz-)scalar σ meson and the (Lorentz-)vector ω meson were considered in order to model thelong-range attraction and short-range repulsion of thenuclear interaction, respectively, in symmetric nuclearmatter. Isovector mesons were added for the descriptionof neutron-proton asymmetric systems. In most models,the vector isovector ρ meson is considered, but anisospin-dependent splitting of the neutron and protonDirac effective masses is obtained only when a scalarisovector δ meson is included. In contrast, the Landaueffective masses are different in asymmetric matter evenwithout a δ meson. Pseudoscalar mesons, such as thepion, or pseudovector mesons are relevant in models thattreat exchange effects explicitly (Lalazissis et al., 2009).The mesons in the QHD approach share the samequantum numbers with their counterparts observed inexperiments; however, they have to be seen as effectivefields in the model that serve to capture the essentialfeatures of the strong interaction in the medium. With thestandard choice of mesons, the contribution of freemesons to the Lagrangian (20) reads

Lmes ¼ 12ð∂μσ∂μσ −m2

σσ2Þ

þ 12ð∂μ

~δ · ∂μ~δ −m2δ~δ · ~δÞ

− 14GμνGμν þ 1

2m2

ωωμωμ

− 14~Hμν · ~H

μν þ 12m2

ρ~ρμ · ~ρμ ð22Þ



with the usual field tensors of the vector mesons Gμν ¼∂μων − ∂νωμ and ~Hμν ¼ ∂μ~ρν − ∂ν~ρμ. In most ap-proaches, it is assumed that mesons couple minimally tonucleons leading to an interaction contribution of the form

Lint ¼ −Xi¼n;p

ψ i½γμðgωωμ þ ~τ · gρρμÞ

þgσσ þ gδ~τ · ~δ�ψ i; ð23Þwhere gi (i ¼ ω, σ, ρ, δ) denote the empirical couplingconstants. Their values are obtained by fitting to propertiesof nuclear matter or finite nuclei. The coupling to scalarmesons modifies the Dirac effective mass m�

i of thenucleons. It is essential in order to obtain a realisticspin-orbit splitting in nuclei. From the Lagrangian (20)the field equations for nucleons and mesons are derived.They have to be solved self-consistently, usually in theRMF approximation, where meson fields are treated asclassical fields and negative-energy states of the nucleonsare neglected (no-sea approximation). Scalar and vectordensities of nucleons appear as source terms for themesons. Finally, a covariant energy density functional isobtained.

The basic version of QHD as discussed previously canqualitatively describe the feature of saturation in nuclearmatter. It results from a competition of attractive scalar andrepulsive vector self-energies Si and Vi in the relativisticdispersion relation

Ei ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þ ðm�

i Þ2q

þ Vi ð24Þ

for a nucleon iwithmomentump andDirac effective massm�

i ¼ mi − Si. With increasing density of the medium, thescalar potential Si rises more slowly than the vectorpotential Vi. The EoS becomes very stiff; nevertheless,the speed of sound does not exceed the speed of light. Aquantitative description of nuclear matter and nucleirequires the extension of the simple Lagrangian density(20) in order to simulate a medium-dependent effectiveinteraction. Several options have been explored in theliterature.

In early extensions of the model, nonlinear (NL) self-interactions of the mesons were considered by adding acontribution of the form

Lnl ¼ −A3σ3 −

B4σ4 þ C

4ðωμω

μÞ2 ð25Þ

to Eq. (20). Cubic and quartic terms of the σ meson wereintroduced by Boguta and Bodmer (1977) and satisfactoryresults were obtained; see, e.g., Reinhard et al. (1986) andRufa et al. (1988). The addition of the quartic ω term bySugahara and Toki (1994)wasmotivated by comparison ofthe scalar and vector potentials with DBHF. Later, self-couplings of isoscalar and isovector mesons were used tomodify the isospin dependence of the EoS (Müller andSerot, 1996; Furnstahl, Serot, and Tang, 1997; Todd-Ruteland Piekarewicz, 2005).

Instead of adding explicit new terms to the Lagrangiandensity (20), the coupling constants gi in Eq. (23) can be

replaced by functionals Γi of the nucleon fields. It is foundthat effective density-dependent (DD) nucleon-mesoncouplings can be extracted from the medium-dependentDBHF nucleon self-energies. Usually, a dependence of thecouplings Γi on the vector density5 nv ¼

ffiffiffiffiffiffiffiffijμjμ

pis

assumed, which is defined in a covariant way with thenucleon current jμ. In the rest frame of a nucleus or nuclearmatter,nv is identical to the baryon number density nB. Thedensity dependence of the couplings leads to so-called“rearrangement” contributions to the vector self-energies(Fuchs, Lenske, and Wolter, 1995). This is essential inorder to obtain a thermodynamic consistent model. Thefunctional formfor the density dependenceof the couplingswith rational and exponential functionswas suggested by acomparison to DBHF results in an early parametrization ofthe DD-RMF model that was fitted to binding energies ofnuclei (Typel and Wolter, 1999). This approach wasfollowed by several others (Nikšić et al., 2002, 2005;Long et al., 2004; Typel, 2005, 2010; Roca-Maza, Viñaset al., 2011). Alternative functions were considered byGögelein et al. (2008). The density dependence of themeson-nucleon couplings was also directly derived fromthe nucleon self-energies (Hofmann, Keil, and Lenske,2001a, 2001b; Gögelein et al., 2008), where the momen-tum dependence of the self-energies was mapped to aneffective density dependence.

Details on applications of the NL and DD-RMFmodels can be found in Reinhard et al. (1986), Rufa et al.(1988), Reinhard (1989), Ring (1996), Serot and Walecka(1997), Bender, Heenen, and Reinhard (2003), Furnstahl(2004), Vretenar et al. (2005), and Nikšić, Vretenar,and Ring (2011). The models are employed withoutexplicitly taking the antisymmetrization of the many-bodywave function into account. The finite range of the effectiveinteraction mediated by the meson exchange requiresextra computational efforts in order to handle fully anti-symmetrized many-body states similar to nonrelativisticGogny HF calculations. In addition, an entirely newparameter set for the couplings has to be determined.Nevertheless, relativistic Hartree-Fock or Hartree-Fock–Bogoliubov models were implemented (Long, Van Giai,and Meng, 2006; Meng et al., 2006; Long et al., 2007,2010).

Treating the basic scalar-vector models as relativisticlocal quantum field theories, it was found that two-loopcorrections lead to large contributions. It was concludedthat the loop expansion does not converge and that thecomposite nature of the nucleons has to be respected byintroducing nonlocal interactions as represented by formfactors; see, e.g., Prakash, Ellis, and Kapusta (1992) andreferences therein.

Somegeneralizationsof theRMFmodel extend the formof the nucleon-meson interaction from minimal couplingsto couplings of the meson fields to derivatives of thenucleon fields. An early version is the model by ZimanyiandMoszkowski (1990), where scalar derivative couplingswere introduced that could be transformed to particular

5Note that despite the name, the vector density is a Lorentz scalar.



nonlinear couplings of nucleons to the σ meson. Generalfirst-order derivative couplings were considered by Typel,von Chossy, and Wolter (2003) and Typel (2005). Theapproach was extended to derivatives of arbitrary highorder by Gaitanos, Kaskulov, and Mosel (2009) andGaitanos and Kaskulov (2012, 2013) in the nonlinearderivative (NLD) coupling model and studied in variousversions (Chen, 2012, 2014; Antić and Typel, 2015;Gaitanos and Kaskulov, 2015). The main feature of theNLD approach is the dependence of the nucleon self-energies not only on the density but also on the energy ormomentum as in DBHF calculations. As a consequence,themomentumdependence of the nucleon optical potentialin nuclear matter, which is extracted from the fitting ofproton-nucleus scattering data in Dirac phenomenology(Hama et al., 1990; Cooper et al., 1993), can be reproducedup to nucleon energies of about 1 GeV.

The explicit appearance of meson fields in the Lagran-gian density of RMF approaches is suppressed in so-called relativistic point-coupling (PC) models (Nikolaus,Hoch, and Madland, 1992; Rusnak and Furnstahl, 1997;Bürvenich et al., 2002; Zhao et al., 2010). Here four-nucleon contact terms, including powers and derivativesthereof, appear with free prefactors that need to bedetermined. They can be seen as the result of expressingsolutions of the meson field equations in the QHDapproach as functions of the relativistic source densitiesand their spatial derivatives. This resembles the nonrela-tivistic SkyrmeHF approach. A systematic expansion ofLin various densities, currents, and their derivatives ispossible by using power counting arguments from princi-ples of effective field theory (Furnstahl, 2002). The formand parameters of the PC approach can also be constrainedby in-medium χEFT (Finelli et al., 2003, 2004, 2006).

Nuclear matter characteristics of 263 RMF parametri-zations were compared recently by Dutra et al. (2014).Similar to the case of SkyrmeHFmodels, only a very smallnumber is consistent with all nuclear matter constraintsconsidered in that publication.

• Quark-meson coupling model: An approach closelyrelated to the previously discussed relativistic descrip-tions of matter and nuclei is the quark-meson coupling(QMC) model (Downum et al., 2006; Rikovska-Stoneet al., 2007; Thomas et al., 2013; Whittenbury et al.,2014). It explicitly considers nucleons as bound states ofquarks which couple to mesons in the surroundingmedium. This leads to a polarization of the nucleonand the resulting mass shift is calculated self-consistently. It can be expressed as a polynomial inthe σ meson field similar to NL-RMF models. Propertiesof matter are calculated in the HF approximation,including pions in addition to the standard scalar andvector mesons. With a small number of parameters,results of similar quality as in (non)relativistic mean-field models or EDFs are obtained.

• Other approaches to the nuclear energy density func-tional: Besides the models discussed previously thatdominate the applications to the EoS, a number ofindependent alternative approaches were developed inthe past. Instead of nonrelativistic effective interactions

of the Skyrme or Gogny type, other forms were inves-tigated, e.g., a density-dependent separable monopoleinteraction (Rikovska Stone et al., 2002) or three-rangeYukawa (M3Y) type interactions (Nakada, 2003). Thephenomenologically inspired approach of Fayans(1998), Fayans et al. (2000), and Fayans and Zawischa(2001) exploits the quasiparticle concept of Migdal’stheory of finite Fermi systems. The Barcelona-Catania-Paris (-Madrid) (BCP or BCPM) EDFs (Baldo et al.,2004, 2013; Baldo, Schuck, and Viñas, 2008) areconstructed by interpolating between parametrizationsof BHF results, obtained with realistic nucleon-nucleonpotentials, for the EoS of symmetric nuclear matter andneutron matter. By adding appropriate surface and spin-orbit contributions, an excellent description of finitenuclei is obtained with only a small number ofparameters.

b. Quark matter

A proper QCD-based description of strongly interactingmatter, in particular, in the vicinity of the predicted deconfine-ment phase transition, is desirable but currently not availablesince the theory is challenging to solve at finite chemicalpotentials. With a few exceptions, the prevailing approach forthe hadron-quark transition region is to describe both phasesseparately and to interpolate in between in terms of a phasetransition construction (see Sec. III.D). Therefore, QM in thefollowing has to be understood as deconfined quark matter. Inthis phase one can think of quarks as actual particles orquasiparticles with no particularly complicated behavior orconfinement properties. Then it is not surprising that typicalapproaches to describe quark matter show many similarities toRMF models for nuclear matter. To model quark mattercorrectly it is important though to understand and to accountfor the confinement mechanism in order to eventually under-stand the phase diagram of strongly interacting matter in thelanguage of QCD. Therefore we briefly review correspondingdevelopments as far as they concern the EoS of dense matter.

• Thermodynamic bag model: The simplest, but stillwidely applied model for QM is a limiting case of theMIT bag model (Chodos et al., 1974), which wasoriginally developed to describe hadrons as quark boundstates of finite size. Confinement in this model isaccomplished by endowing the finite region with aconstant energy per unit volume B. A special case isa highly excited hadron in which quarks would thenbehave as an ideal gas. The latter idea was followed todescribe a system of homogeneous, deconfined quarkmatter (Farhi and Jaffe, 1984). The EoS is that of an idealFermi gas of three quark flavors, where the bag constantB is added to the total energy density and subtractedfrom the pressure in order to maintain thermodynamicconsistency. B can be understood as the pressure differ-ence of confined and deconfined quarks in vacuum. Thevalue of B can be determined from more sophisticatedmodels (Cahill and Roberts, 1985). The bag constantarises not solely due to deconfinement but rather fromthe breaking of chiral symmetry. Consequently it isdensity dependent (Buballa and Oertel, 1999) as well as



flavor dependent (Buballa, 2005). Already in perturba-tive QCD the thermodynamic bag models treatment ofquarks as free noninteracting fermions does not hold andrequires corrections (Freedman and McLerran, 1977c;Fraga, Pisarski, and Schaffner-Bielich, 2001), which canbe generalized to a simple power series expansion ofthe pressure in the quark chemical potential (Alfordet al., 2005). Similarly, a phenomenologically motivatedexpansion considers diquark contributions to the pres-sure (Alford and Reddy, 2003). Recently, an extensionof the thermodynamic bag model was suggestedwhich accounts for the breaking of chiral symmetryand the influence of vector interactions (Klähn andFischer, 2015).

• Nambu–Jona-Lasinio–type models and extensions: Oneof the prominent features of QCD is the dynamicalbreaking of chiral symmetry as the mechanism thatgenerates most of the hadron masses. The idea thatthe nucleon mass can be understood as the self-energy ofa fermion in analogy to the energy gap of a super-conductor has been developed at a time where the notionof quarks did not even exist (Nambu and Jona-Lasinio,1961a, 1961b). The NJL model is based on a Lagrangianfor a fermion field with a quartic, chirally symmetriclocal interaction of the form

L ¼ ψðγμi∂μ −mÞψ þ GfðψψÞ2 þ ðψiγ5~τψÞ2g; ð26Þ

where ψ is understood as a quark fermion spinor,m is thebare quark mass, G is a coupling constant, and ~τ are theisospin matrices. The similarity to RMF models isevident. A Fierz transformation of the interaction givesaccess to all possible quark-antiquark interaction chan-nels. These techniques have been explained in detail byKlevansky (1992) and Buballa (2005) for vacuum and in-medium applications. The coupling G can be understoodas a particular choice of a generalized form factorfor nonlocal current-current interactions (Schmidt,Blaschke, and Kalinovsky, 1994; Bowler and Birse,1995). An important feature of the approach is thatthe quartic terms—as a hallmark of NJL-type models—can be shown to bosonize into baryon, meson, anddiquark contributions to the partition function and henceto the thermodynamic potential (Kleinert, 1976; Roberts,Cahill, and Praschifka, 1988; Cahill, Praschifka, andBurden, 1989; Hatsuda and Kunihiro, 1994). Thisbosonization property is exploited to develop an under-standing of hadrons as quark bound states in the medium(Bentz and Thomas, 2001; Wang, Wang, and Rischke,2011; Blaschke et al., 2014). The original NJL model’ssuccess rests on its ability to describe the breaking ofchiral symmetry. It fails to describe the infrared behaviorof QCD which addresses confinement. It has thereforebeen suggested to extend the model by an imaginarychemical potential expressed in terms of the Polyakovloop as a possible order parameter of deconfinement(Fukushima, 2004; Ratti, Thaler, and Weise, 2006). Amodification of the Polyakov loop potential in thesePNJL models due to a finite quark chemical potential μ

was introduced by Dexheimer and Schramm (2010).Further extensions result from variations of the previousmodels, e.g., nonlocal PNJL models without (Blaschkeet al., 2008) and with (Contrera, Grunfeld, and Blaschke,2014) μ-dependent Polyakov loop potential. The PNJLmodel is used to study the QCD phase diagram at finitetemperatures and densities (Fukushima, 2008). A furtherapproach to account for confinement in NJL-type modelssuggests to introduce an infrared cutoff to removeunphysical quark-antiquark thresholds (Ebert, Feldmann,and Reinhardt, 1996). The similarity of the NJL model tothe RMF approach for nuclear matter suggests that inanalogy to the nonlinear Walecka model higher-ordercoupling channels, i.e., multiquark or quark-mesoninteractions, will affect, in particular, the high-densitybehavior of the EoS (Benic et al., 2015; Zacchi, Stiele,and Schaffner-Bielich, 2015).

C. Clustered and nonuniform matter

At subsaturation densities and not too high temperatures,nucleonic matter is no longer uniform since it becomesunstable with respect to variations in the particle densities.There are various criteria to identify the onset of instabilities,in both static and dynamic approaches; see Sec. III.D fordetails. Spatial structures can develop on different lengthscales. In stellar matter, nucleons can form nuclei or clusters ofdifferent sizes and shapes due to the interplay between theshort-range nuclear interaction and the long-range electro-magnetic interaction. If the size of the clusters is small ascompared to their mean free path, the matter can still bedescribed as a homogeneous system, however with clusterdegrees of freedom in addition to nucleons and leptons. Athigher densities, e.g., in the so-called pasta phases in the innercrust of NSs, the density variations have to be treatedexplicitly. In low-density cold matter, nuclei arrange them-selves in a lattice and a crystal structure develops, e.g., in theouter crust of NSs, and a new length scale emerges. Theappearance of cluster structures in matter can be treated invarious approximations that differ mainly with respect to thechoice of the basic degrees of freedom and to the descriptionof interactions.

1. Nuclear statistical equilibrium

Themost basic approach to describe clusteredmatter is givenby NSE models, which are sometimes just called “statisticalmodels.” They are characterized by assuming a statisticalensemble of different nuclear species and nucleons in thermo-dynamic equilibrium. In particular, the chemical potentials ofnuclei are not independent. They are given by Eq. (2). NSEmodels are not only used in simulations of CCSNe but also fornucleosynthesis calculations and in the context of thermonu-clear supernovae; see, e.g., Seitenzahl et al. (2009).In its simplest form, the ideal NSE, a mixture of

noninteracting ideal gases assuming Maxwell-Boltzmannstatistics is utilized. In chemistry, this description is knownas “mass-action law.” In the ideal NSE approach the abun-dance ratio of nuclei is determined by a Saha equation, whichoriginally was used to describe the population of ionization



states in atoms. Instead of classical Maxwell-Boltzmannstatistics, the correct quantum statistics of the particles canalso be implemented. Often the corresponding Fermi-Diracdistribution is adopted only for nucleons.Standard NSE models do not take into account the effects

of the strong interaction between the constituents explicitly,e.g., correlations in nucleon-nucleon scattering states areneglected. However, in some models the interactions in thenucleonic component are incorporated by employing a mean-field description of homogeneous matter; see Sec. III.B.2.a.A large variety of NSE based models can be found in the

literature, employing various extensions and different levels ofsophistication. Here we summarize only the most importantaspects which are typically discussed in the context of thesemodels. For some selected models we give further detailsin Sec. V.

a. Nuclear binding energies

Realistic EoSs in the NSEmodel description require nuclearbinding energies as basic input. Different approaches are used:On the one hand, values from theoretical models are employed.These can be simple mass formulas such as liquid-drop-likeparametrizations or more detailed nuclear structure calcula-tions; see, e.g.,Myers and Swiatecki (1990, 1994),Möller et al.(1995), Lalazissis, Raman, and Ring (1999), Geng, Toki, andMeng (2005), and Koura et al. (2005). On the other hand,experimentallymeasured binding energies (Audi,Wapstra, andThibault, 2003; Wang et al., 2012) are used directly. However,an extension of these mass tables from experiments is requiredwith the help of theoretical approaches since exotic nuclei thatare not yet studied experimentally can be encountered. Shelleffects in the structure of nuclei have a significant impact on thedistribution at low temperatures as illustrated in Sec. V.C.Binding energies of nuclei inside matter are modified ascompared with their vacuum values. These medium effectsare often introduced in NSE models in phenomenologicalapproximations (see Sec. III.C.1.d).

b. Excited states

It is straightforward to include excited states of nuclei in anexplicit way if their excitation energies are known exper-imentally. However, especially at high excitation energies andfor very heavy or exotic nuclei, the experimental informationon the levels and their properties is not complete. In thiscase, theoretical level densities or internal partition functionscan be used (Fái and Randrup, 1982; Engelbrecht andEngelbrecht, 1991; Iljinov et al., 1992; Blinnikov et al.,2011). Alternatively, a temperature dependence of the bindingenergies is introduced (Botvina and Mishustin, 2010). Theworks of Rauscher and Thielemann (2000, 2001), andRauscher (2003) provide nuclear partition functions in tabularform for temperatures up to 24 MeV and a wide range ofnuclear masses. These calculations are based on both exper-imental data and a backshifted Fermi-gas model. Morerecently, similar tables were provided by Goriely, Hilaire,and Koning (2008). Problems with divergences of the originalFermi-gas model (Bethe, 1936) at low excitation energy canbe solved (Grossjean and Feldmeier, 1985). However, thegeneral reliability of the employed densities of state formulas

can be questioned. For an investigation of effects of excitedstates on the supernova EoS and the stellar collapse, see, e.g.,Mazurek, Lattimer, and Brown (1979), Nadyozhin and Yudin(2004), and Liu, Zhang, and Luo (2007).

c. Coulomb interaction

In matter with the condition of electric charge neutrality, theCoulomb interaction among nucleons and nuclei is screeneddue to the background of electrons and possibly muons. SomeNSE models neglect this Coulomb screening. Others include itby using only the one-body Wigner-Seitz approximation(Lattimer et al., 1985). However, there are very detailedcalculations available that were obtained from the study ofsingle-component and even multicomponent plasmas at differ-ent temperatures; see, e.g., Chabrier and Potekhin (1998),Chugunov and DeWitt (2009), Potekhin et al. (2009, 2013),and Potekhin, Chabrier, and Rogers (2009) and Sec. III.D.2for more details. Typically the results are provided in the formof fitting formulas. The simulations have reached a highnumerical precision and deviations from the linear mixing rulefor binary plasmas were found to be small (DeWitt, Slattery,and Chabrier, 1996). For a discussion of different approx-imations of Coulomb interactions in supernova EoS and theapplication of some of the aforementioned models, see, e.g.,Nadyozhin and Yudin (2005) and Blinnikov et al. (2011).

d. Medium modifications of heavy nuclei

Some statistical models employ explicit medium correctionsof the binding energies of heavy nuclei. These can be due totemperature or due to the presence of unbound nucleons. Inboth cases, the surface and bulk properties of nuclei aremodified as compared to the vacuum at zero temperature.One aspect is a temperature dependence of the symmetryenergy (Dean, Langanke, and Sampaio, 2002; Agrawal et al.,2014) and of effective nucleon masses (Donati et al., 1994;Fantina et al., 2012). Obviously, such temperature effects arerelated to excited states and internal partition functions but theproblem is approached from a different perspective. Effects ofthe unbound nucleons on nuclei are often extracted fromnucleons-in-cell calculations (see Sec. III.C.6). For instance,Papakonstantinou et al. (2013) and Aymard, Gulminelli, andMargueron (2014) calculated the binding energy shifts forSkyrme interactions in the local-density approximation and inthe extended Thomas-Fermi (TF) approximation, respectively.Itwas pointed out that the definition of the binding energy shiftshas to be consistentwith the definition of clusterswhere one hasto distinguish coordinate-space and energy-space clusters.

e. Cluster dissolution

The application of the standard NSE is limited to rather lowdensities. This model cannot describe the dissolution of nucleiwith increasing densities, the Mott effect, which is mainlydriven by the Pauli principle (Röpke, Münchow, and Schulz,1982; Röpke et al., 1983). When the nuclear saturation densityis approached, a transition to uniform nucleonic matter is oftenenforced with the help of the excluded-volume mechanism,which represents a classical, phenomenological approach todescribe the dissolution of nuclei at high densities in a



geometrical picture. Different variants of excluded-volumeeffects can be found in the literature. In the simplest case, thetotal volume of the thermodynamic system is replaced by theso-called free volume that is the total volume reduced bythe volume occupied by the particles of finite size (Rischkeet al., 1991). This approach is well known from the EoS of avan der Waals gas. General expressions for excluded-volumeeffects can be found in Yudin (2010, 2011) and Typel (2016).More detailed models solve for the exact canonical partitionfunction, taking into account finite volumes of the particlesand/or assuming a certain geometry of the particles. Animportant example is the hard-sphere model (Carnahan andStarling, 1969; Mulero, 2008). Note that the excluded-volumemechanism is also commonly used in the context of relativisticheavy-ion collisions (HICs) to describe the freeze-out ofparticles and their yields in a hadron resonance gas model(Gorenstein, Petrov, and Zinovjev, 1981; Andronic et al.,2012). A problem of the excluded-volume approach is theoccurrence of a superluminal speed of sound at high densitiesand hence the EoS becomes acausal (Rischke et al., 1991;Venugopalan and Prakash, 1992).

2. Single nucleus approximation

Instead of considering the distribution of all nuclei, thechemical composition is sometimes simplified by assuming arepresentative single heavy nucleus, unbound nucleons, andpossibly α particles and other light nuclei in the description.Burrows and Lattimer (1984) showed that this so-called singlenucleus approximation (SNA) has only a small impact onthermodynamic quantities. However, there can be significantdifferences between the average mass and charge number ofheavy nuclei in a full NSE model and the correspondingvalues of the representative nucleus employing the SNA(Souza et al., 2009) (see also Sec. V.C). Furthermore, theconclusions of Burrows and Lattimer (1984) are not appli-cable if the composition is dominated by light nuclei. In thiscase it is not possible to consider a “representative lightnucleus” due to the small number of nuclei involved and thelarge variability of their binding energies (Hempel et al., 2012,2015). We point out that considering a statistical ensemble ofall nuclei, i.e., going beyond the SNA, is particularly relevantfor the determination of electron-capture rates during corecollapse (see Sec. VI.B.1).

3. Virial expansion

Correlations between the constituents of a low-density gasof particles at finite temperature can be considered in the virialequation of state (VEoS). It provides corrections to the NSEapproach to clustered matter. The original formulation goesback to Beth and Uhlenbeck (1936, 1937) and uses adescription based on the grand canonical ensemble. TheVEoS relies on a series expansion of the grand canonicalpotential

ΩðT; V; fμigÞ ¼ −TV�X

i

biλ3i

zi þXij

bij

λ3=2i λ3=2j

zizj þ � � ��

ð27Þ

in powers of the particle fugacities zi ¼ exp ½ðμi −miÞ=T�,where μi is the chemical potential of particle species iincluding the rest mass mi. The quantities λi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2π=miT

pdenote the thermal wavelengths and bi are the degeneracyfactors of the single particles. The virial coefficients bij, bijk,etc. are simple functions of the temperature. They containinformation on the two-, three-, etc., many-body correlationsin the system. In particular, the second virial coefficients

bij ¼λ3=2i λ3=2j

2λ3ij

ZdE exp

�−ET

�DijðEÞ ð28Þ

with λij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2π=ðmi þmjÞT

pdepend only on the energies

EðijÞl;k of the two-particle bound states and the two-body

scattering phase shifts δðijÞl in channels l that appear in thefunction

DijðEÞ ¼Xl

�Xk

gðijÞl;k δðE − EðijÞl;k Þ þ

gðijÞl

π

dδðijÞl

dE

�ð29Þ

with appropriate degeneracy factors gðijÞl;k and gðijÞl . Note thatthe separation of bound state (first term) and scatteringcontributions (second term) is not unique, since a partialintegration of the energy integral (28) leads to equivalentexpressions with a different partitioning. Because experimen-tal information can be used directly in the evaluation, a model-independent approach is obtained. Quantum statistical (QS)effects can be incorporated easily. For contributions beyondsecond order see, e.g., Pais and Uhlenbeck (1959), Dashen,Ma, and Bernstein (1969), Bedaque and Rupak (2003), andLiu, Hu, and Drummond (2009). The VEoS is only applicablefor small fugacities zi ≪ 1 or equivalently niλ3i ≪ 1 with theparticle number density ni. These thermodynamic conditionsare found, e.g., in the neutrino sphere of supernovae. Inpresent applications of the VEoS to stellar matter onlycorrelations on the level of the second virial coefficient dueto the strong interaction are considered.

4. Quantum statistical approach

As mentioned, the transition from inhomogeneous to uni-form matter cannot be easily described within the NSEapproach nor within the VEoS. For that purpose, NSE modelsare extended phenomenologically by adding a classicalexcluded-volume correction (see Sec. III.C.1.e). A systematicdescription of correlations and, in particular, the Mott effect inan interacting many-body system is given by the QS approach(Röpke, Münchow, and Schulz, 1982; Röpke et al., 1983).In general, many-body methods can provide spectral

functions that contain all information on correlations.Prominent peaks in a spectral function can be identified withthe corresponding quasiparticles, e.g., deuterons in nucleonicmatter. As an approximation, quasiparticles with shiftedenergies can be introduced in the practical calculation of anEoS. These energies depend on the nucleon densities, thetemperature, and the momentum of the quasiparticle in themedium. They include effects of Pauli blocking and can beparametrized with more or less sophistication (Röpke, 2009,



2011; Typel et al., 2010). The energy shifts enter in thedetermination of the particle densities and finally in the EoS.Going beyond the simple quasiparticle approximation for

the spectral function in the QS approach leads to a form of thegrand canonical potential that closely resembles the corre-sponding expression of the VEoS. This generalized Beth-Uhlenbeck approach was formulated and studied in detail withrealistic two-nucleon potentials (Schmidt, Röpke, and Schulz,1990; Stein, Morawetz, and Röpke, 1997). In this case, theEoS can be formulated with a modified expression forthe second virial coefficient. The quasiparticle energies inthe bound state contribution are affected by the in-mediumenergy shifts and the phase shifts in the scattering contributionhave to be calculated from the in-medium T matrix for two-body scattering.

5. Generalized relativistic density functional

The modification of cluster binding energies in the medium,which is a basic feature of the QS approach, can beimplemented in quasiparticle models. The generalized rela-tivistic density functional (gRDF) approach (Typel et al.,2010, 2014; Typel, 2013) is such a model. It is an extension ofa RMF model with density-dependent nucleon-mesoncouplings that is constrained by fits to properties of finitenuclei. In the gRDF model, nucleons and clusters are includedas explicit degrees of freedom. Two-nucleon scatteringcorrelations are described by effective resonances in thecontinuum with parameters that are adjusted in order toreproduce the model-independent virial EoS at low densities(Voskresenskaya and Typel, 2012). All particles couple to themeson fields with appropriately scaled strengths resulting indensity-dependent scalar and vector self-energies. In addition,mass shifts of composite particles are implemented in aparametrized form, which are derived from the QS approach,in order to account for Pauli blocking from the nucleons in themedium. A further change of the cluster binding energies iscaused by the screening of the Coulomb field due to theelectronic environment in stellar matter. The density depend-ence of the binding energy shifts and meson-nucleoncouplings leads to rearrangement contributions in the vectorself-energy and thermodynamic quantities, which guarantee athermodynamically consistent approach. The model interpo-lates between the correct low-density limit given by the virialEoS and the suprasaturation case of purely nucleonicmatter. Medium-dependent mass shifts of light clusters werealso incorporated in other models that study pasta phases incompact star matter (Avancini et al., 2012; Pais, Chiacchiera,and Providência, 2015).

6. Nucleons-in-cell calculations

In all models that were previously described, matter istreated as a uniform system of interacting particles. They areassumed to be pointlike or to have a finite volume. Theformation of inhomogeneous structures in subsaturationmatter at low temperatures can be studied in calculationswhere a nonuniform distribution of nucleons inside a cell ofgiven shape and size is considered. Here quantal and classicalmethods can be distinguished. In stellar matter, electrons areusually not treated explicitly but considered as a uniform

background gas. The methods are mostly applied to study theformation of clusters or pasta phases in the crust of neutronstars, the corresponding phase transitions, and the effects ofthe Coulomb interaction (see Sec. III.D). Beyond staticproperties for an EoS, the dynamical response, e.g., inneutrino scattering, or hydrodynamic quantities such asviscosities and conductivities can also be studied.In the classical molecular dynamics (CMD) approach

(Dorso, Molinelli, and Lopez, 2011; Horowitz et al., 2011;Dorso, Giménez Molinelli, Nichols, and Lopez, 2012;Piekarewicz and Toledo Sanchez, 2012; Schneider et al.,2013; Giménez Molinelli et al., 2014; Giménez Molinelliand Dorso, 2015), a certain number of nucleons is placedinside a box of given volume to reproduce a fixed density. Thevelocities of the particles are chosen to represent a Maxwelliandistribution at given temperature. The particles interact via atwo-body potential where the Coulomb contribution is essen-tial. The time evolution of the system is followed by solving aset of classical coupled equations of motion, which is possibleeven for a rather large number of particles. In quantummolecular dynamics (QMD) models (Maruyama et al.,1998; Kido et al., 2000; Watanabe et al., 2002a, 2002b,2003a, 2004, 2009; Matsuzaki, 2006; Watanabe, 2007;Sonoda et al., 2008; Maruyama, Watanabe, and Chiba,2012), particles are not treated as pointlike objects as inCMD calculations but as wave packets of Gaussian shape. Themotion of the centroid of the wave package is classicallyfollowed. Quantum statistical effects, such as antisymmetri-zation or shell effects, are not accounted for in moleculardynamics (MD) simulations. The Pauli exclusion principlecan be incorporated approximately with appropriatelydesigned contributions to the potentials. Central questionsof MD calculations are the fragment recognition (Dorso andRandrup, 1993; Strachan and Dorso, 1997), the chemicalcomposition (Horowitz, Berry, and Brown, 2007; Horowitzand Berry, 2009; Dorso, Giménez Molinelli, López, andRamirez-Homs, 2012; Caplan et al., 2014), the appearanceof different, sometimes complicated, shapes in the densitydistribution and their topological characterization (Watanabeet al., 2003b; Dorso, Giménez Molinelli, and López, 2012;Alcain, Giménez Molinelli, Nichols, and Dorso, 2014;Schneider et al., 2014; Horowitz et al., 2015), and phasetransitions (Watanabe et al., 2005; Alcain, Giménez Molinelli,and Dorso, 2014). Structure functions and quantities related tothe dynamical response can be extracted as well (Horowitzet al., 2004, 2005; Horowitz, Perez-Garcia, and Piekarewicz,2004; Caballero et al., 2008; Horowitz and Berry,2008, 2009).A description of matter inside a cell based on particle

density distributions instead of localized classical particles is awidely adopted approach for inhomogeneous systems. Anumber of different methods is available. They differ in thephysical input, the approximations, and the numerical com-plexity. Many approaches that model the formation of largenuclei surrounded by a gas of nucleons employ the Wigner-Seitz approximation where the size of the cell is determined bythe neutrality condition, i.e., the total charge of baryonicmatter inside the volume considered is compensated by theelectronic charge. At low densities, a spherical cell, which iscentered around individual nuclei, is usually assumed. When



other geometries, mostly rectangular shapes, are considered,it is common to presume a periodic density distributionin space.The (compressible) liquid-drop model (LDM) belongs to

the class of microscopic-macroscopic approaches where theenergy of matter inside the cell is parametrized withseveral individual contributions such as bulk, surface, etc.(Baym, Bethe, and Pethick, 1971; Lattimer, 1981; Lamb et al.,1983; Ravenhall, Pethick, and Wilson, 1983; Lattimer et al.,1985; Lattimer and Swesty, 1991; Lorenz, Ravenhall, andPethick, 1993; Watanabe, Iida, and Sato, 2000; Douchin andHaensel, 2001; Oyamatsu and Iida, 2007; Nakazato,Oyamatsu, and Yamada, 2009; Furusawa et al., 2011;Nakazato, Iida, and Oyamatsu, 2011; Furusawa, Sumiyoshiet al., 2013). They depend on the size of the heavy nucleus andthe density of the surrounding gas of nucleons. The detailedstructure of the expressions can be guided by energy densityfunctionals, often nonrelativistic Skyrme parametrizations.Because of its convenience in the numerical application,the LDM is one of the earliest approaches to model inhomo-geneous stellar matter.Other widely employed methods are the TF approximation

or extensions thereof. Here the density distribution of fer-mions in the cell is determined using a given EDF fornucleonic matter. In the most simple form, a local-densityapproximation of nuclear matter is obtained, but correctionsrelated to surface effects and the finite range of interactions,can be incorporated in the calculation. The shape of thedensity distributions can be given by a simple functional formwith few parameters or can be determined fully self-consis-tently. Technically, the free energy of the cell is minimizedvariationally and the chemical potentials of the nucleons areobtained for fixed particle numbers. Both nonrelativistic andrelativistic (Sumiyoshi, Oyamatsu, and Toki, 1995; Cheng,Yao, and Dai, 1997; Shen et al., 1998a, 2011; Avancini et al.,2008, 2009; Avancini, Barros et al., 2010; Avancini,Chiacchiera et al., 2010; Okamoto et al., 2012; Zhang andShen, 2014) models are available. The main disadvantage ofthe TF approach is that shell effects are not included. They canbe incorporated with the Strutinski method (Brack, Jennings,and Chu, 1976; Brack, Guet, and Hakansson, 1985; Onsiet al., 2008; Pearson et al., 2012, 2015) but this correction tothe TF approach is not widely used for astrophysical EoS. Theformation of inhomogeneous matter inside cells of variousshapes is also studied by using the density functional theory(DFT) formalism in relativistic approaches (Maruyama et al.,2005; Gögelein and Müther, 2007), where shell effects areincluded but only in the mean-field approximation withoutexplicit particle exchange.The next level of complexity is reached in HF calculations,

which take the antisymmetrization of the nuclear many-body wave function within the cell fully into account. Thisapproach is most frequently utilized with nonrelativisticpotentials such as the zero-range Skyrme interaction, e.g.,in the pioneering works of Ravenhall, Bennett, and Pethick(1972), Negele and Vautherin (1973), Bonche and Vautherin(1981, 1982), and Bonche, Levit, and Vautherin (1985). Shelleffects were found to disappear quickly with increasingtemperature at 2–3 MeV and nucleon pairing effects arerelevant only below approximately 1 MeV (Brack and

Quentin, 1974a, 1974b). The evolution of the density dis-tribution inside the cells and consequences for the EoS areinvestigated in several works; see, e.g., Sil et al. (2002),Magierski, Bulgac, and Heenen (2003), Gögelein and Müther(2007), Gusakov, Kantor, and Haensel (2009a), Newton andStone (2009), Pais and Stone (2012), Papakonstantinou et al.(2013), and Pais, Newton, and Stone (2014). Also time-dependent HF calculations were employed, both in thestandard formalism (Schuetrumpf et al., 2013a, 2013b,2015) and in an extended approach using dynamical waveletsfor single-particle wave functions (Sebille, Figerou, and de laMota, 2009; Sebille, de la Mota, and Figerou, 2011). A correcttreatment of antisymmetrization beyond the individual cellvolume was developed recently (Vantournhout, Jachowicz,and Ryckebusch, 2011; Vantournhout et al., 2011;Vantournhout and Feldmeier, 2012).

D. Phase transitions

Theoretical models for the EoS are mostly concerned withcalculations of single phases where a thermodynamic poten-tial is extremized locally for its set of natural state variables(see Sec. II.A). In the case of thermodynamic instabilities, theequilibrium state of the system is obtained from a globalminimization or maximization of the appropriate thermody-namic potential allowing for a phase transition with coexist-ence of different phases, i.e., macroscopic regions in spacewith different values of the various densities but identicalintensive variables (Landau and Lifshitz, 1980).Examples of phase transitions in the present context are the

liquid-gas phase transition in nuclear matter (Barranco andBuchler, 1980; Müller and Serot, 1995; Gulminelli et al.,2003; Hempel et al., 2013), the hadron-quark transition,which is expected to occur at very large baryon numberdensities and/or temperatures (Collins and Perry, 1975;Prakash, Cooke, and Lattimer, 1995; Steiner, Prakash, andLattimer, 2000; Mishustin et al., 2002; Bhattacharyya,Mishustin, and Greiner, 2010; Hempel et al., 2013;Yasutake et al., 2013), and the gas–liquid-solid phase tran-sition, which is relevant in the formation of the crystallinecrust during the cooling of PNSs (Chamel and Haensel, 2008).A phase transition could also be caused by the appearance ofnew particle species such as hyperons in dense hadronicmatter (Schaffner-Bielich and Gal, 2000; Schaffner-Bielichet al., 2002; Gulminelli, Raduta, and Oertel, 2012; Gulminelliet al., 2013).

1. Thermodynamic description of phase transitions

In the following, FðT; fNig; VÞ is chosen as the thermo-dynamic potential for the discussion. The free energydescribes the equilibrium thermodynamics of a systemif it is a convex function of the extensive variables, i.e.,fNig and V, and a concave function of the temperature T.Then the conjugate intensive variables, the chemical potentialsμi ¼ ∂F=∂NijT;fNj≠ig;V , and the pressure p ¼ −∂F=∂VjT;fNigare constant throughout the volume V. The free energy of aparticular theoretical model is locally convex in the subspaceof extensive variables if all eigenvalues of the stability matrixM with entries



Mij ¼∂2F

∂Qi∂Qj

��T;Qk;k≠i;k≠j

ð30Þ

are positive. Here Qi, Qj, and Qk are variables from the set ofthe conserved charges fNig or the volume V. If at least oneeigenvalue of M is zero or negative, this point in the space ofvariables is metastable or unstable, respectively. All unstablepoints are enclosed by the so-called spinodal in the space ofconserved charge numbers NB, NQ, etc. Besides this thermo-dynamic criterion there are other dynamical approaches toidentify the instability region, e.g., collective excitations withrandom phase approximation calculations, the Vlasov equa-tion formalism, or Fermi liquid theory (Pethick and Ravenhall,1988; Heiselberg, Pethick, and Ravenhall, 1993; Margueron,Navarro, and Blottiau, 2004; Brito et al., 2006; Providência,Brito, Avancini et al., 2006; Providência, Brito, Santos et al.,2006; Ducoin, Margueron, and Chomaz, 2008; Ducoinet al., 2008).The spinodal is enclosed by the binodal that connects all

points with identical temperature, chemical potentials, andpressure. Points inside the binodal belong to the phasecoexistence region. Here the free energy for given conservedtotal charge numbers can be lowered as compared to thelocally calculated value by considering the coexistence oftwo phases p ¼ I, II with different volumes Vp such thatVI þ VII ¼ V. The particle numbers Np

i in the individualphases are in general different but all chemical potential areidentical μIi ¼ μIIi in the two phases as well as the otherintensive variables (PI ¼ PII, TI ¼ TII). This is called theGibbs condition for thermodynamic equilibrium. Points incoexistence always lie on a binodal. Hence, it is sufficient toknow the thermodynamic potential on this line or surface toconstruct the system properties for thermodynamic conditionsinside the binodal.If there is only a single conserved charge, it is easily

checked whether the free energy of a particular theoreticalmodel is a convex function of the only conserved charge forgiven T and V. If not, the binodal degenerates to two separatepoints and the well-known Maxwell construction of phasetransitions with isotherms in the pressure-density diagram isobtained. By changing the temperature of the system, the twodistinct points in coexistence can collapse into a single pointthat defines the critical point and the corresponding criticaltemperature and critical pressure. If this topology applies, it ispossible to move from one phase to the other around thecritical point without crossing the binodal in the temperature-density plane. In some cases, e.g., for the quark-hadron phasetransition, there is often no consistent model for the completerange of the thermodynamic variables and two differentmodels are used. Then a transition from one phase to theother without crossing a phase separation line is not possible(Hempel et al., 2013).In the case of more than one conserved charge the complete

set of Gibbs conditions applies and the topology of thebinodals and spinodals becomes more complex (Barrancoand Buchler, 1980; Glendenning, 1992; Müller and Serot,1995; Iosilevskiy, 2010; Hempel et al., 2013). New featuresappear: In the higher dimensional parameter space, the criticalpoint turns into a critical line or critical hypersurface and

several topological end points can be defined. However, it isalways possible to map the Gibbs construction with severalindependent charge numbers to a technically more simpleMaxwell construction (Ducoin, Chomaz, and Gulminelli,2006; Typel et al., 2014). This is achieved by applyingLegendre transformations to the free energy F that replaceall conserved charge numbers except one with the correspond-ing chemical potentials. Thereby a modified thermodynamicpotential is found that depends only on a single-particlenumber as in the standard Maxwell case.

2. Coulomb effects

In models of the EoS for NSs and CCSNe the equilibriumwith respect to the strong as well as the electromagneticinteraction between all constituents has to be considered.As a consequence, the phase structure of dense matter issubstantially affected by the interplay of these short- and long-range forces in competition with entropy. In addition, thespecific condition of charge neutrality applies. At not toohigh temperatures the appearance of clusters and crystal-line structures is expected (see also Sec. III.C). Closelyconnected to both features is the possible occurrence ofpasta phases.If macroscopic phases coexist, different assumptions for the

treatment of charge neutrality can be made. Either one requireslocal charge neutrality, i.e., each phase is charge neutral, orthe system is charge neutral as a whole and the phases areallowed to carry a net charge. We note that the assumption ofcharged phases but global charge neutrality contradicts theassumption of the thermodynamic limit, if interpreted strictly,as the Coulomb energy would diverge for such a system.Nevertheless, it is considered as a reasonable simplification inmany situations; see Martin and Urban (2015) for a discussionregarding the inner NS crust. Depending on the choice forrealizing the charge neutrality condition, the chemical equi-librium conditions for phase coexistence (Hempel, Pagliara,and Schaffner-Bielich, 2009) lead to different qualitativeproperties of the phase transition (Glendenning, 1992;Iosilevskiy, 2010; Hempel et al., 2013), in particular, to thefeature of Coulomb frustration (Gulminelli et al., 2003;Chomaz et al., 2007; Napolitani et al., 2007; Hasnaoui andPiekarewicz, 2013).In more advanced approaches, surface effects and the finite-

range of interactions are explicitly taken into account. Acrucial ingredient is the surface tension between phases.Typically, for high surface tensions, the phases tend toapproach a configuration that resembles the case of localcharge neutrality (Heiselberg, Pethick, and Staubo, 1993;Maruyama et al., 2008; Yasutake et al., 2013). For very lowsurface tensions, however, the configuration can be similar tothe case of global charge neutrality without finite-size effects.The liquid-gas phase transition in nuclear matter predicts

the coexistence of high-density and low-density phases ofmacroscopic size below a critical temperature. If Coulombeffects are included, electrons have to be added to compensatethe positive proton charge. Phase transitions with macro-scopic, charge-neutral phases in coexistence would createlarge electric fields at the interfaces that the system tries toavoid (Voskresensky, Yasuhira, and Tatsumi, 2003). This can



be seen in the analysis of instabilities where density fluctua-tions of certain wavelengths are preferred. Clusters of finitesize emerge that are surrounded by a low-density gas ofnucleons. The density of the electrons throughout the systemis nearly constant due to the large incompressibility of such ahigh-density Fermi liquid. Often a constant electron density isassumed, with some exceptions; see, e.g., Maruyama et al.(2005), Endo et al. (2006), and Yasutake, Maruyama, andTatsumi (2011). The formation of clusters in uniform nuclearmatter is well studied in several models that consider thedistribution of nucleons and electrons in a cell of givengeometry (see Sec. III.C.6). Usually, the Wigner-Seitzapproximation is applied. At low temperatures and ratherlow densities, a single cluster in a spherical cell is the preferredgeometry (Lamb et al., 1978; Douchin and Haensel, 2000).With increasing density it becomes more advantageous todevelop structures of cylindrical or planar geometry and asequence of pasta phases is found; see, e.g., Watanabe and Iida(2003) and Gupta and Arumugam (2013) and references givenin Secs. III.C.6 and V.A.1. This was observed in early modelsfor cells with different symmetry with simple energy densityfunctionals (Hashimoto, Seki, and Yamada, 1984; Oyamatsu,Hashimoto, and Yamada, 1984; Williams and Koonin, 1985;Oyamatsu, 1993). In principle, every change from onegeometry to another represents a phase transition of itsown. More refined calculations with less restrictions to thespatial distributions of the densities found that transitions fromclustered matter at low densities to uniform matter at highdensities exhibit weaker discontinuities, i.e., the phase tran-sitions are much less pronounced (Newton and Stone, 2009;Pais, Newton, and Stone, 2014). A nearly complete quenchingof the traditional liquid-gas phase transition can occur(Gulminelli et al., 2003).When stellar matter is cooled down at a given density, the

size of clusters grows as observed, e.g., in spherical Wigner-Seitz cell calculations. At very low densities, the averagedistance between the clusters is large and the short-rangestrong interaction can practically be neglected. Coulomb andthermal energies drive the thermodynamic behavior of thesystem that can be seen as a plasma of ions and electrons. Atvery low temperatures a phase transition from the gas phase toa crystalline phase is expected (Salpeter, 1961; Baym, Pethick,and Sutherland, 1971). The ratio of the strengths of theCoulomb interaction and of the thermal energy is measured bythe parameter

Γi ¼Z5=3i e2

aeTð31Þ

with the charge Zi of the cluster and the electronic length scale

ae ¼�3ne4π

�1=3

ð32Þ

depending on the electron density ne. When the temperatureapproaches zero, Γi diverges. Fully ionized electron-ionplasmas have been studied in detail (Brush, Sahlin, andTeller, 1966; Hansen, 1973; Pollock and Hansen, 1973;Farouki and Hamaguchi, 1993; Chabrier and Potekhin,1998; Potekhin and Chabrier, 2000, 2010; Chabrier,

Douchin, and Potekhin, 2002; Daligault, 2006; Cooper andBildsten, 2008; Potekhin et al., 2009; Potekhin, Chabrier, andRogers, 2009). From classical Monte Carlo simulations (seeSec. III.C.6) of a one-component plasma (OCP) it is knownthat the phase transition from the gas to the crystal phaseoccurs at Γi ≈ 175. The exact value will depend on the detailsof the theoretical model. In particular, the often employedWigner-Seitz approximation is insufficient to capture thecorrelations that are induced by the Coulomb interactionand determine the location of the phase transition. For amixture of different ionic species, corrections to the OCPresult apply (Ogata et al., 1993; Chabrier and Potekhin, 1998;Nadyozhin and Yudin, 2005; Chugunov and DeWitt, 2009).For Γi → 0 the Debye screening limit in a plasma is obtained.For Γi → ∞ a body-centered cubic (bcc) lattice of ionsimmersed in a uniform sea of electrons is found as the groundstate (Baym, Bethe, and Pethick, 1971; Baym, Pethick, andSutherland, 1971). At finite temperatures lattice vibrationscontribute to the thermodynamic potential of the system(Baiko, Potekhin, and Yakovlev, 2001). With increasingtemperature these thermal excitations will lead to the meltingof the crystal.An amorphous structure instead of a crystal as a ground

state at zero temperature seems to be unlikely but the actualion lattice type could be very sensitive to detailed conditions(Ichimaru, Iyetomi, and Mitake, 1983; Magierski and Heenen,2002). However, Jog and Smith (1982) found that a phasecomposed of interpenetrating cubic lattices of different nucleican be preferred in certain density regions.

IV. CONSTRAINTS ON THE EoS

Models for the EoS can be constrained by differentobservables, which originate mainly from three differentsources as follows: (1) laboratory measurements of nuclearproperties and reactions, (2) theoretical ab initio calculations,and (3) observations in astronomy. These constraints can testdifferent regions in the space of thermodynamic variables.They are, in the best case, independent of each other anddifferent aspects of an EoS model can be checked; see, e.g.,Klähn et al. (2006), Lattimer and Prakash (2007), Tsang et al.(2012), Lattimer and Lim (2013), Li and Han (2013),Horowitz et al. (2014), Lattimer and Steiner (2014), andStone, Stone, and Moszkowski (2014) for discussions.Although we discuss a fair amount of constraints, keep inmind that these are usually limited to very restricted domainsin the phase diagram (e.g., saturation properties are propertiesof symmetric matter, NSs are cold and do not explicitly probethe EoS at given density, etc.). Therefore, theoretical modelsare required to interpolate between or even extrapolateaway from these constrained regions. While it is desirablethat these models by themselves do not add further uncer-tainties one has to be cautiously aware that this is notnecessarily the case.Properties of nuclear matter are usually characterized by a

number of parameters that are related to the leading contri-butions in an expansion of the energy per nucleon

EðnB; δÞ ¼ E0ðnBÞ þ EsymðnBÞδ2 þOðδ4Þ ð33Þ



in the isospin asymmetry δ ¼ 1 − 2Yq. Both the energy pernucleon of symmetric matter

E0ðnBÞ ¼ mnuc − Bsat þ 12Kx2 þ 1

6Qx3 þ � � � ð34Þ

and the symmetry energy

EsymðnBÞ ¼ J þ Lxþ 12Ksymx2 þ � � � ð35Þ

can be expanded close to nuclear saturation in the deviationx ¼ ðnB − nsatÞ=3nsat of the baryon density nB from thesaturation density nsat. Equations (34) and (35) define thebinding energy at saturation Bsat, the incompressibility K,the skewness Q, the symmetry energy at saturation J, thesymmetry energy slope parameter L, and the symmetryincompressibility Ksym. In general, nuclear matter parametersare strongly correlated among each other as well as toproperties of nuclei and neutron stars; see, e.g., Klüpfel et al.(2009), Kortelainen et al. (2010), Lattimer and Lim (2013),and Lattimer and Steiner (2014). In recent years, many studiesfocused on obtaining constraints for the symmetry energyEsym and its density dependence; see, e.g., the contributions tothe topical issue by Li et al. (2014). The most importantconstraints on the nuclear matter parameters and the EoS arediscussed in more detail in the following sections.

A. Terrestrial experiments

1. Systematics from nuclear masses and excitations

The most basic and least ambiguous constraints for the EoScome from properties of nuclei, most notably nuclear masses(Audi, Wapstra, and Thibault, 2003; Wang et al., 2012) anddensity distributions (De Vries, De Jager, and De Vries, 1987;Angeli and Marinova, 2013). An extrapolation to infinitemass numbers yields the corresponding nuclear matter param-eters. Besides the saturation point at saturation density ofnsat ≈ 0.15–0.16 fm−3, and the corresponding value of thebinding energy of Bsat ≈ 16 MeV, accurate constraints on thesymmetry energy Esym and its density dependence areobtained. There exists a linear correlation between thesymmetry energy at saturation J and the slope parameter L(Lattimer and Lim, 2013; Lattimer and Steiner, 2014). Thiscorrelation is very robust and validated in various theoreticalapproaches; see, e.g., Kortelainen et al. (2010), Fattoyev andPiekarewicz (2011), and Nazarewicz et al. (2014).The aforementioned correlation is based on ground-state

binding energies. Instead of ground-state binding energies,Danielewicz and Lee (2014) considered excitation energies toisobaric analog states and charge invariance to derive con-straints for the symmetry energy. In a comprehensive analysis,Skyrme HF calculations were used to derive an acceptableregion for Esym at densities from 0.04 to 0.16 fm−3. They alsoextracted a constraint for J and L which significantly overlapswith the constraint from nuclear masses. At baryon densitiesnB ∼ 0.105 fm−3, the constraint of Danielewicz and Lee(2014) is the tightest, with an excellent accuracy of�1.2 MeV. For higher densities the constraint rapidly dete-riorates, for lower densities it gets slightly worse. This“bottleneck” region was previously noted by Brown (2000),

Trippa, Colò, and Vigezzi (2008), and Roca-Maza, Brennaet al. (2013). Nuclear energy density functionals with differentvalues of J and L that are fitted to binding energies of nucleioften show a crossing of their symmetry energies and/or theirneutron matter EoS in this region. The corresponding densitycan be interpreted as an average value of the densities in finitenuclei. Danielewicz and Lee (2014) combined their analysisof isobaric analog states with measurements of skin thick-nesses to arrive at tighter constraints of J ¼ 30.2–33.7 MeVand L ¼ 35–70 MeV.Obviously, constraints on the symmetry energy become

tighter, if the experimental knowledge about binding energiesis extended to very asymmetric nuclei. Many of the currenthigh-precision mass measurements have been made possibleby Penning-trap mass spectrometers or mass spectrometrywith storage rings in combination with radioactive beams(Wolf et al., 2013). Binding energies of nuclei are also crucialfor nucleosynthesis calculations and the location of the driplines (Erler et al., 2012). They can also be used directly in theEoS of the outer crust of cold NSs; see, e.g., Baym, Pethick,and Sutherland (1971), Kreim et al. (2013), and Wolf et al.(2013), and Sec. V.A.1.

2. Nuclear resonances

Nuclear resonances in the form of collective excitations offinite nuclei contain important information about the isoscalarand isovector properties of the nucleon-nucleon interaction.For example, Paar et al. (2014) performed a global statisticalanalysis of experimental results for different collective exci-tations with emphasis on correlations between differentobservables. In addition to nuclear masses and charge radii,they considered the anti-analog giant dipole resonance, theisovector giant quadrupole resonance, the dipole polarizabilityof 208Pb, and the pygmy dipole resonance transition strengthin 68Ni. Employing a certain class of relativistic nuclear EDFsleads to tight constraints for J ¼ 32.5� 0.5 MeV and L ¼49.9� 4.7 MeV and the crust-core transition density in NSs.Interestingly, the former values are fully compatible withthe final results of Lattimer and Lim (2013) with J ¼29.0–32.7 MeV and L ¼ 40.5–61.9 MeV and Lattimer andSteiner (2014) with L ¼ 44–66 MeV.

a. Giant monopole resonance

Constraints for the nuclear incompressibility K can bededuced from fitting results of theoretical models to exper-imental data on the isoscalar giant monopole resonance(ISGMR), also called the breathing mode. However, it isperceived in the literature that the extraction of K fromISGMR data is not unambiguous as it relates to the densitydependence of the symmetry energy in the models(Piekarewicz, 2004; Shlomo, Kolomietz, and Colò, 2006;Sharma, 2009). For example, RMF models often obtain largervalues for K in the range of 250–270 MeV (Piekarewicz,2004) than nonrelativistic models.Recently, Khan and Margueron (2013) reanalyzed the

problem of model dependencies. They showed that the dataactually constrain the density-dependent incompressibilityaround the crossing density of 0.1 fm−3, by using bothrelativistic and nonrelativistic EDFs. Therefore constraints



on K depend also on the skewness parameter Q of thefunctional used to analyze the data. The situation is similarfor the extraction of the symmetry energy from nuclear masses(cf. Sec. IV.A.1).A very comprehensive list of theoretical calculations of K

from the literature was given by Stone, Stone, andMoszkowski (2014). In the same article, a reanalysis of theISGMR was performed, based on a liquid-drop approach tothe description of the vibrating nucleus. Interestingly, it wasfound that K lies in the range of 250–315 MeV, which issignificantly higher than the generally accepted values of K ¼248� 8 MeV (Piekarewicz, 2004) or K ¼ 240� 20 MeV(Shlomo, Kolomietz, and Colò, 2006). They achieved con-sistency with the latter values provided the ratio of the surfaceto volume contributions Ksurf=Kvol in a leptodermous expan-sion is close to −1, as predicted by a majority of mean-fieldmodels. However, in their analysis it seems that the exper-imental data favor a ratio different from −1. The high values ofK are thus related to a different surface contribution to theISGMR compared with other works employing mean-fieldmodels. Note that Stone, Stone, and Moszkowski (2014) wereable to explain the ISGMR of tin isotopes, which was foundby Piekarewicz (2010) to be a startling problem of nuclearstructure.

b. Giant dipole resonance

The nuclear isovector giant dipole resonance (IVGDR)can be used to constrain the symmetry energy. Frommeasured centroid energies for a liquid droplet model oneobtains a correlation between the volume and surface partof the symmetry energy of finite nuclei (Lipparini andStringari, 1989; Lattimer and Lim, 2013), which can betransformed into a correlation between L and J. Trippa,Colò, and Vigezzi (2008) found that the IVGDR gives thetightest constraints on Esym around nB ¼ 0.1 fm−3 with23.3 MeV < Esymð0.1 fm−3Þ < 24.9 MeV, by analyzing theIVGDR by a variety of Skyrme models. Lattimer and Lim(2013) used different functional forms of Esym to extract thecorrelation between L and J. Their results show a significantoverlap with other constraints; see Fig. 1 in Lattimer andSteiner (2014).The pygmy dipole resonance (PDR) at excitation energies

much below the IVGDR is also sensitive to the symmetryenergy (Klimkiewicz et al., 2007; Carbone et al., 2010).Reinhard and Nazarewicz (2010) and Daoutidis and Goriely(2011) argued that it is not possible to extract constraints on Jand L from PDR strengths because their correlation with thesymmetry energy is too weak. This was studied in more detailby Reinhard and Nazarewicz (2013) who found that thecorrelation between the accumulated low-energy strength andthe symmetry energy “dramatically depends on the energycutoff” used. Furthermore, they came to the conclusion thatthe low-energy dipole excitations cannot be interpreted interms of a collective PDR mode.

c. Electric dipole polarizability

Tamii et al. (2011) reported a precise measurementof the electric dipole response of 208Pb from proton inelastic

scattering. The extracted electric dipole polarizabilityαD is correlated to the neutron skin thickness (Lipparini andStringari, 1989; Reinhard and Nazarewicz, 2010; Piekarewiczet al., 2012), which in turn is correlated with L (seeSec. IV.A.3). Using this two-step process, Lattimer andLim (2013) obtained an anticorrelation between L and J,with significant overlap with other constraints.Recently Roca-Maza, Centelles et al. (2013) found that the

product αDJ is much better correlated with the neutron skinthickness of 208Pb and L than the polarizability αD itself.After reanalyzing the experimental results of Tamii et al.(2011), Roca-Maza, Centelles et al. (2013) obtained alinear correlation between J and L. Adopting a value of J ¼31 � 2 MeV, this resulted in L ¼ 43� ð6Þexpt � ð8Þtheor �ð12Þest MeV, where “expt” denotes experimental, and “theor”theoretical uncertainties, while “est” originates from the uncer-tainty in J. Tamii, von Neumann-Cosel, and Poltoratska (2014)obtained a linear correlation between J andL. They pointed outthat the difference to the anticorrelation found by Lattimer andLim (2013) results from how they analyzed the data. Lattimerand Steiner (2014) revised the results of Lattimer and Lim(2013), taking the improved correlation of Roca-Maza,Centelles et al. (2013) into account. Roca-Maza et al. (2015)confirmed the correlation with additional data on other nuclei,slightly enlarging the interval for J and L.Zhang and Chen (2015) analyzed the data from Tamii et al.

(2011) in yet another way. Instead of constraining nuclearmatter properties at normal nuclear density, they showed thatαD of 208Pb puts stringent constraints on the symmetry energy,or almost equivalently the pure neutron matter EoS, atsubsaturation densities significantly below nsat. Their finalresults for the subsaturation EoS are consistent with theexperimental constraints of Tsang et al. (2009) andDanielewicz and Lee (2014). In addition, they obtainedagreement with various theoretical works for the neutronmatter EoS, which are included in Fig. 6. The recent study ofHashimoto et al. (2015) determined the dipole polarizabilityof 120Sn, which is strongly correlated with that of 208Pb,experimentally from proton inelastic scattering.

3. Neutron skin thicknesses

The density distributions of nucleons and their root-mean-square (rms) radii

ffiffiffiffiffiffiffiffihr2i i

pchange rather smoothly for nuclei in

the valley of stability when the mass number increases.However, the proton and neutron radii are not in generalequal. Neutron-rich nuclei develop a neutron skin with

thickness Δrnp ¼ffiffiffiffiffiffiffiffihr2ni

p−

ffiffiffiffiffiffiffiffiffihr2pi

q. The charge distributions

and charge radii of many nuclei are well known experimen-tally, e.g., from elastic electron scattering or isotope shiftmeasurements; see Angeli et al. (2009) and Angeli andMarinova (2013) and references therein. In contrast, neutronradii of nuclei and thus neutron skin thicknesses are much lessprecisely determined.For the measurement of neutron radii of nuclei, experiments

with particles that probe the neutron distribution with the helpof the strong or weak interaction have to be utilized. Typicalexamples are proton scattering experiments (Ray, 1979; Rayand Hodgson, 1979; Klos et al., 2007; Terashima et al., 2008;



Zenihiro et al., 2010), isovector giant dipole excitationsby inelastic α-particle scattering (Krasznahorkay et al.,1994), ð3He; tÞ charge exchange reactions (Krasznahorkayet al., 1999), the excitation of pygmy dipole resonances(Klimkiewicz et al., 2007), or the study of antiprotonic atoms(Trzcinska et al., 2001; Jastrzebski et al., 2004; Brown et al.,2007). Parity violation in elastic electron scattering is used inthe lead radius experiment PREX at Jefferson Lab (Horowitzet al., 2001, 2012; Abrahamyan et al., 2012). In this type ofapproach the weak form factor of the nucleus is measured andit is primarily determined by the neutron density distribution.Unfortunately, the deduced neutron skin thickness of 0.302�0.175ðexpÞ � 0.026ðmodelÞ � 0.005ðstrangeÞ fm carries alarge uncertainty. It is expected to diminish in future exper-imental runs. A noticeably smaller value of Δrnp ¼ 0.15�0.03ðstatÞþ0.01

−0.03 ðsysÞ fm was recently reported from experi-ments of coherent pion photoproduction at the MAMI electronfacility (Tarbert et al., 2014).Brown (2000) found a strong correlation of the neutron skin

thickness of 208Pb with the derivative dEðnB; δÞ=dnBjnB¼n0;δ¼1

of the neutron matter EoS at a density n0 ¼ 0.1 fm−3 innonrelativistic HF calculations with 18 different parametriza-tions of the Skyrme interaction. These observations triggeredmany theoretical and experimental studies to explore therelation of isospin-dependent properties of nuclei to theEoS, in particular, the density dependence of the nuclearsymmetry energy EsymðnÞ. An extension of the Skyrme HFcalculations in similar studies of RMF models (Typel andBrown, 2001), general density functionals in the context ofEFT (Furnstahl, 2002) or the droplet model (Warda et al.,2009) showed the same correlation, which can also beexpressed as a correlation between Δrnp and the slopeparameter L. More recent representations of the Δrnp-Land similar correlations of isospin-dependent properties canbe found in Centelles et al. (2009), Chen et al. (2010), Roca-Maza, Centelles et al. (2011), Gaidarov et al. (2012, 2014),Tsang et al. (2012), and Viñas et al. (2014a, 2014b). Theconsequences for the properties of NSs, such as radius orproton fraction, were studied by Horowitz and Piekarewicz(2001a, 2001b, 2002), Steiner et al. (2005), Todd-Rutel andPiekarewicz (2005), and Avancini et al. (2007a, 2007b,2007c). The origin of the Δrnp-L correlation, its bulk, andsurface contributions in nuclei and the relation to Landau-Migdal parameters have been discussed by Dieperink et al.(2003), Centelles et al. (2010), and Warda et al. (2010). Theneutron skin thickness provides a correlation between L and Jas well and shows a decreasing of L with increasing J, incontrast to other correlations of that type (Lattimer and Lim,2013). The present data of the Δrnp-L correlation are derivedonly from mean-field calculations of nuclear structure.However, the neutron skin thickness could be modified bynucleon-nucleon correlations and clustering at the surface ofthe nucleus (Typel, 2014).

4. Heavy-ion collisions

The EoS of warm or hot, strongly interacting matter can beconstrained in laboratory experiments with HICs. Dependingon the beam energy, the impact parameter, the choice of

observables, and the combination of projectile and targetnuclei, very different conditions can be explored. In the earlyphase of almost central collisions of about 1 GeV, highdensities of up to 4 times the nuclear saturation densityand temperatures of about 40–50 MeV can be reached for avery short time (Blättel, Koch, and Mosel, 1993; Fuchs et al.,1997). In the later stages of a collision, more peripheral or lessenergetic reactions, properties of dilute matter at temperaturesbelow the critical temperature of the liquid-gas phase transition(15–20 MeV) and subsaturation densities can be studied.We will not consider here ultrarelativistic HICs probing matterat very low baryon density and high temperatures. Since thephysics of HICs is a large field on its own, we mention onlythe most important aspects relevant to this review.There are fundamental differences between matter in HICs

and in compact stars. Temperatures and densities can besimilar to those in CCSNe, but matter in HICs is usually moreisospin symmetric (cf. Sec. II.B.3). Furthermore, the fireball ina HIC has a finite size with a fixed number of nucleons that arenot necessarily in thermal equilibrium. This limits, forexample, the maximum mass number of nuclear clustersformed under these conditions. Matter in compact stars,which can be treated in the thermodynamic limit, has to becharge neutral, whereas there is a net charge in HICs fixedby the initial charge of the two colliding nuclei. In HICsCoulomb interactions are typically neglected because of thehigh kinetic energies. Characteristic time scales in HICs areof the order of a few fm=c and do not allow for equilibriumwith respect to weak interactions. On the contrary, incatalyzed NSs full equilibrium is reached. In compact stars,weak equilibrium with respect to strangeness changingreactions is usually assumed, whereas the net strangenessin HICs is zero. These differences have to be taken intoaccount when comparing astrophysical EoSs with con-straints from HICs.The analysis of HICs requires the comparison of measured

data to rather complex theoretical simulations since a dynami-cal process has to be followed. These models are based ondifferent approaches that aim to solve the relevant transportequations. On the one hand, a set of Boltzmann-type equationsfor the single quasiparticle distribution functions is consid-ered; see, e.g., Danielewicz (1984a, 1984b). They can bederived consistently as an approximation of the nonequili-brium Kadanoff-Baym theory including collision and some-times fluctuation terms (Bertsch and Das Gupta, 1988; Busset al., 2012). On the other hand, simulations with moleculardynamics models in classical approximations, possiblyincluding antisymmetrization effects, are also employed(Aichelin, 1991; Ono et al., 1992; Hartnack et al., 1998).One major challenge is to predict the distribution of observedparticles and fragments reliably. In these models the EoS doesnot directly enter but the interactions between all particleswhich participate in the collision, as well as in-medium crosssections of the relevant reactions, which are usually para-metrized in a convenient form (Li and Chen, 2005). Oneimportant aspect is the momentum dependence of the inter-action because particle momenta attain much larger values inHICs than in nuclei (Chen et al., 2014; Xu, Chen, and Li,2015). Hydrodynamic descriptions (Welke et al., 1988; Galeet al., 1990; Huovinen and Ruuskanen, 2006; Gale, Jeon, and



Schenke, 2013), which can make direct use of an EoS, aremore appropriate for studying the evolution of the high-density phase of a collision, in particular, in (ultra)relativisticHICs. However, one has to change to a different approach atlater times when the system expands, the density drops, andfragments are formed.There are several observables in HICs that are sensitive to

particular features of in-medium interactions that determinethe EoS at suprasaturation densities (Fuchs and Wolter, 2006).Not only nucleons but also mesons, such as pions or kaons, aswell as light nuclei, e.g., 2H, 3H, 3He, and 4He [see, e.g.,Chajecki et al. (2014)], are valuable messengers for theproperties of the medium at high and low densities,respectively.The collective flow of nucleons exhibits a distinct azimuthal

distribution (Welke et al., 1988), which can be characterizedwith coefficients in a Fourier analysis. The transverse flow inperipheral reactions seems to be mainly sensitive to themomentum dependence of the mean field. The elliptic flow,in contrast, depends strongly on the maximum compressionthat is reached and it is correlated with the stiffness of the EoS.The analysis of laboratory experiments in comparison withsimulations indicates that the incompressibility of symmetricnuclear matter cannot be too high (Welke et al., 1988;Danielewicz, Lacey, and Lynch, 2002; Reisdorf et al.,2012; Le Fèvre et al., 2016) (see Sec. V.D.3 and Fig. 16).The collision region with the highest densities is best

studied with particles that are produced only there and interactweakly with the medium after their formation. Although beinga rare probe due to their subthreshold production (Hartnacket al., 2012), kaons seem to be a good choice. Theirobservation in HICs points toward a rather low incompress-ibility K with values below 250 MeV (Fuchs et al., 2001;Sturm et al., 2001; Hartnack, Oeschler, and Aichelin, 2006).Consequences of these constraints from HICs on compact starproperties were explored by Sagert, Tolos et al. (2012). For adiscussion of the interplay between HICs and astrophysicaldata see, e.g., Aichelin and Schaffner-Bielich (2009).In recent years, HIC experiments for constraining the EoS

mainly focused on the isospin dynamics (Li, 2002; Baranet al., 2005; Li, Chen, and Ko, 2008; Di Toro et al., 2009;Wolter et al., 2009; Tsang et al., 2012; Cozma et al., 2013;Ademard et al., 2014; De Filippo and Pagano, 2014; Kohleyand Yennello, 2014) in order to explore the properties ofisospin asymmetric matter in more detail. Yield ratios ofparticle pairs with the same mass but different isospin, such asn=p, πþ=π− (Xiao et al., 2014), or fragments 3H=3He (Yonget al., 2009) have been intensively investigated. The consid-eration of single or double ratios has the advantage thatsystematic experimental uncertainties are reduced and thesensitivity is increased. A possibility to amplify isospin-dependent effects is the comparison of collisions with differ-ent combinations of projectiles and targets with more or lessneutron excess. For example, the isospin diffusion in the neckregion of peripheral and midcentral collisions of 112Sn=124Snnuclei is sensitive to the symmetry potential (Tsang et al.,2009, 2012).The density dependence of the symmetry energy at mod-

erate to high densities was studied with the help of n=p ratios

and their elliptic flow difference (Cozma, 2011; Russottoet al., 2011, 2013, 2014) as well as πþ=π− ratios (Reisdorfet al., 2007). The analysis of the latter results within transportmodel simulations suggests a decrease of the symmetryenergy at high densities, which is in conflict with mostEoS models (Xiao et al., 2009; Xie et al., 2013). Also, thepuzzling results for the effective mass splittings of nucleons inintermediate HICs (Zhang et al., 2014; Coupland et al., 2016)still need a satisfactory explanation (Kong et al., 2015). Onlymore accurate measurements will allow one to set tighterbounds on the symmetry energy at high densities.Multifragmentation reactions probe conditions very similar

to matter in CCSNe as illustrated in Fig. 4, where typicalconditions for CCSNe and multifragmentation reactions areindicated; see the caption for details. In these reactions, athermalized system of nuclear matter is formed that ischaracterized by subnuclear densities and temperatures of3–8 MeV. The deexcitation of the system occurs via nuclearmultifragmentation, i.e., breakup into many excited fragmentsand nucleons. For the theoretical description of such reactions,for instance the statistical multifragmentation model (SMM) isused that is presented in more detail in Sec. V.C. Statisticalmodels accurately describe many characteristics of the nuclearfragments observed in the experiments: cluster multiplicities,charge and isotope distributions, various correlations, and

FIG. 4. Nuclear phase diagram in the temperature–baryondensity plane. Solid and dash-dotted blue lines indicate bounda-ries of the liquid-gas coexistence region for symmetric andasymmetric matter calculated with TM1 interactions (Sugaharaand Toki, 1994). The shaded area corresponds to typical con-ditions for nuclear multifragmentation reactions (Botvina andMishustin, 2010). The dashed black lines are isentropic trajecto-ries characterized by constant entropy per baryon, s ¼ 1, 2, 4, and6 calculated with the statistical model for supernova matter(SMSM) (Botvina and Mishustin, 2010). The dotted red linesshow results of a CCSN simulation from Sumiyoshi et al. (2005)just before bounce (BB), at core bounce (CB), and postbounce(PB). From Buyukcizmeci et al., 2013.



other observables; see, e.g., Gross (1990), Bondorf et al.(1995), and Botvina and Mishustin (2010).The observation of light nuclei, which are emitted in

Fermi-energy HICs, allows one to determine the densityand temperature of warm dilute matter from experiments(Kowalski et al., 2007; Natowitz et al., 2010; Wada et al.,2012). The derived symmetry energies of the clustered matterindicate an increase as compared to those obtained in modelcalculations of uniform matter that is assumed to be composedsolely of nucleons. The thermodynamic conditions are similarto those in the neutrinosphere of CCSNe (Horowitz et al.,2014). From the observed yields of nucleons and clusters, itwas possible to extract the in-medium binding energies andMott points of light clusters (Hagel et al., 2012) with the helpof chemical equilibrium constants (Qin et al., 2012). Hempelet al. (2015) refined the study of Qin et al. (2012), takinginto account the differences between matter in HIC andCCSNe. Results for the equilibrium constant of the α particleare presented in Fig. 5. A comparison of many EoSs forwarm dilute matter shows that simple NSE descriptions arenot sufficient to reproduce experimental data (Hempelet al., 2015).

B. Neutron matter calculations

The simple isospin structure of pure neutron matter sim-plifies the nuclear interaction Hamiltonian, such that ab initiocalculations can be carried out more easily than in the case ofgeneral asymmetric nuclear matter. Calculations using thedifferent many-body techniques introduced in Sec. III.B withwell-calibrated interactions are available for a large range in

densities; see, e.g., the review by Gandolfi, Gezerlis, andCarlson (2015) and references therein. They can serve asimportant constraints for the EoS models, discussed in Sec. V,although their results are not directly applicable to astro-physical objects.At very low densities, neutron matter is dominated by s-

wave interactions with a large scattering length a ¼ −18.5 fm,indicating that the two-neutron system is almost bound.Neutron matter at these densities is close to the unitary limit,explored experimentally for cold fermionic atoms (Ho andMueller, 2004; Ho and Zahariev, 2004).At intermediate densities, up to roughly nuclear matter

saturation density, and at higher densities, relevant for NSs,many different calculations from ab initio methods exist. Wemention some calculations, without claiming completenessfor the list below. Seminal results are the variational calcu-lations by Friedman and Pandharipande (1981) and Akmal,Pandharipande, and Ravenhall (1998) using the Urbana andArgonne nuclear two- and three-body forces. BHF calcula-tions have been reported for instance by Baldo, Bombaci, andBurgio (1997) and Zhou et al. (2004), and DBHF results byvan Dalen, Fuchs, and Faessler (2004), Krastev andSammarruca (2006), and Sammarruca et al. (2012). Rios,Polls, and Vidaña (2009) compared SCGF calculations forthermodynamic properties of hot neutron matter with thecorresponding BHF calculations. Carbone (2014) obtainedSCGF results including an effective three-nucleon force forfinite temperature and extrapolated them to vanishing temper-ature. Horowitz and Schwenk (2006b, 2006c) applied thevirial expansion to dilute neutron matter at nonzero temper-ature. An early application of the virial expansion was thedescription of a neutron gas in supernovae by Buchler andCoon (1977) using the soft-core Reid potential. QMC calcu-lations for zero temperature neutron matter using differentversions of the Argonne and Urbana nuclear potentials havebeen presented by Carlson et al. (2003), Gandolfi et al.(2009), Wlazłowski and Magierski (2011), and Gandolfi,Carlson, and Reddy (2012). Recent QMC (Gezerlis et al.,2013, 2014; Roggero, Mukherjee, and Pederiva, 2014;Wlazłowski et al., 2014) and coupled cluster (Baardsen et al.,2013; Hagen et al., 2014) calculations employ chiral poten-tials. Neutron matter is particularly interesting for chiralforces, since only a few LECs accompanying the contactterms are involved up to next-to-next-to-next-to leading order(N3LO) including three- and four-nucleon forces. In addition,at least up to roughly saturation density, the MBPT resultswith RG evolved and unevolved chiral forces are in very goodagreement, showing that neutron matter in this range behavesperturbatively to a very good approximation; see, e.g., Tolos,Friman, and Schwenk (2008) and Krüger et al. (2013).Comparison with QMC calculations corroborates the pertur-bative nature of neutron matter at these densities (Gezerliset al., 2014). Calculations of neutron matter with chiral forcescan be found in Hebeler and Schwenk (2010, 2014), Krügeret al. (2013), and Tews et al. (2013). In-medium χEFTfollowing different power counting schemes was applied toneutron matter at zero and nonzero temperature by Lacour,Oller, and Meißner (2011), Fiorilla, Kaiser, and Weise (2012),and Drischler, Soma, and Schwenk (2014).

104

105

106

107

108

109

1010

1011

Kc[

](fm

9 )

4 5 6 7 8 9 10 11 12 13 14

T (MeV)

Exp. (Qin et al. 2012)ideal gasHS(DD2), no CS, A 4SFHo, no CS, A 4LS220, HIC mod., cor. BSTOS, HIC mod.SHT(NL3)SHO(FSU2.1)gRDFQSFYSS, A 4, no CS

FIG. 5. Equilibrium constants of α particles extracted from HICexperiments (black diamonds) in comparison with those ofvarious theoretical models, which are all adapted for the con-ditions in HICs, as far as possible. The gray band is theexperimental uncertainty in the temperature determination. Theblack line shows the equilibrium constant of the ideal gas model.From Hempel et al., 2015.



In Fig. 6 we show the energy per baryon of pure neutronmatter as a function of baryon number density as obtained in afew of the different approaches cited previously. The yellowregion corresponds to the AFQMC calculations of Gandolfiet al. (2009) and Gandolfi, Carlson, and Reddy (2012) with theArgonne two-body potential. The width of the band indicatesuncertainties related to the different phenomenological three-body forces. The AFQMC results of Wlazłowski et al. (2014)with chiral forces (green lines and symbols) including onlytwo-nucleon interactions (2NF) and those including a three-nucleon force (2NFþ 3NF) are in good agreement with theformer ones. The variational results of Friedman andPandharipande (1981) gave lower values than Akmal,Pandharipande, and Ravenhall (1998) and lie at the lowerboundary of the AFQMC calculations. The width of MBPTresults by Hebeler et al. (2013) shows mainly uncertainties inthe three-nucleon forces employed. The constraint derived byKrüger et al. (2013) at saturation density comparing differentchiral forces and cutoff schemes within a MBPT calculation isshown as a vertical error bar. Figure 6 also displays the χEFTresults by Fiorilla, Kaiser, and Weise (2012) and BHF calcu-lations from Vidaña et al. (2010). The latter use Argonne plusphenomenological three-body potentials.We address two points concerning these different results.

First, three-nucleon forces (and potentiallymore, depending onthe resolution scale and density) are important in neutronmatterand add substantial repulsion at and above saturation density.This is seen, for example, by comparing QMC calculationswith andwithout three-nucleon forces. Second, all results, withphenomenological or chiral forces, applying different many-body techniques, are in reasonable agreement up to saturationdensity. This shows that the ab initio many-body calculationsrepresent a reliable constraint on the EoS of neutron matter upto nuclear densities (see Sec. V.D.3 and Fig. 16).The situation is different for calculations of symmetric

nuclear matter. They have been a cornerstone for many-bodymethods for decades. However, the empirical saturation pointis difficult to obtain. In addition, symmetric matter is unstablewith respect to cluster formation at densities below saturationwhich is strongly temperature dependent and leads to an

increase of the binding energy. Therefore theoretical many-body calculations of symmetric matter are not as reliable as forneutron matter and cannot serve as a constraint on the EoS atpresent. Instead phenomenological models are adjusted to theempirical properties of symmetric matter.

C. Astrophysical observations

1. Neutron star masses and radii

Presently, the main astrophysical constraint stems from themeasurements of two very massive NSs in NS-white dwarfsystems which have been reported with unprecedented highprecision. For the first binary system, the determination isbased on Shapiro delay, a general relativistic effect (Demorestet al., 2010). It yields a mass of 1.928� 0.017M⊙ (Fonsecaet al., 2016). In the second case a well-known structure modelfor thewhite dwarf is combinedwith the analysis of orbital datato obtain a mass of 2.01� 0.04M⊙ for the NS (Antoniadiset al., 2013). There are indications of evenmoremassiveNSs inblack widow and redback systems (van Kerkwijk, Breton, andKulkarni, 2011; Romani et al., 2012; Kaplan et al., 2013). Inthese cases, the pulsar is accompanied by a low-masscompanion of a few 0.001M⊙ (black widows) or near0.2M⊙ (redbacks), which is bloated and strongly irradiatedby the pulsar. However, the analysis of these systems is muchmore model dependent than for NS-white dwarf systems. Inparticular, the companion’s light curve has to be modeledinducing large uncertainties in the mass determination.Although the most probable mass for the NS indicates a verymassive object, the results do not yet reach the same reliabilityas the mass determinations of Antoniadis et al. (2013) andFonseca et al. (2016). This also holds for theNS in the eclipsingx-ray binary Vela X-1, where a high mass of 2.12� 0.16M⊙has been reported by Falanga et al. (2015).Smaller NS masses have been measured in various binary

systems; see Lattimer (2012) for a recent compilation. In somecases masses have been derived very precisely from the orbitalparameters of the system without much model dependence inthe analysis. Particularly precise measurements have beenperformed for several binary NS systems giving masses closeto the canonical value of 1.4M⊙.At the other end, the lowest NS masses could be interesting

for constraining the EoS via their formation history.Originally, Podsiadlowski et al. (2005) suggested to considerpulsar B in the double pulsar system J0737-3039, with a verylow and precisely measured mass of 1.2489� 0.0007M⊙. If itoriginates from the collapse of a progenitor star withO-Ne-Mg core and the loss of matter during the formationof the NS is negligible, the baryon number, or equivalently thecorresponding baryon mass MB for the NS, is stronglyconstrained from the properties of the white dwarf progenitor.Its mass was determined to be 1.366M⊙ ≤ MB ≤ 1.375M⊙(Podsiadlowski et al., 2005), assuming a stationary non-rotating object. Kitaura, Janka, and Hillebrandt (2006)concluded on a slightly smaller but similar mass of MB ¼1.36� 0.002M⊙ from simulations of an electron-capturesupernova. A similar system J1756-2251 was recentlyobserved with a slightly lower gravitational mass of 1.230�0.007M⊙ for the pulsar with the lower mass (Ferdman et al.,

0

5

10

15

20

25

30

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

E/A

(M

eV)

nB (fm-3)

AFQMC (AV + TBF)AFQMC (2NF)

AFQMC (2NF + 3NF)APR

Friedman-PandharipandeHebeler et al. (2010)

Friedman PandharipandeHebeler et al. (2010)

χPT, Fiorilla et al. (2012)BHF (AV + TBF)

FIG. 6. Comparison of results for the energy per baryon ofneutron matter at T ¼ 0 MeV from different ab initio ap-proaches. The vertical bar represents the range given at saturationdensity in Krüger et al. (2013). For details see text.



2014). The constraint on the EoS arising from the relationbetween gravitational and baryon mass of these low-mass NSsdepends strongly on assumptions. First, there is no completeconsensus about the formation history of these systems andthe origin from a O-Ne-Mg electron-capture supernova is notconfirmed (Tauris et al., 2013). Second, already a possiblebaryon loss of 1% during the formation of the compact starbroadens the corresponding baryon mass region by increasingit by roughly a factor of 2. This effect is included in theconstraints derived by Kitaura, Janka, and Hillebrandt (2006)but only for two particular EoSs.The ultimate constraint on the EoS is a determination of

radius and mass of the same object; see, e.g., Özel and Psaltis(2009), Read et al. (2009), Özel, Baym, and Güver (2010),and Steiner, Lattimer, and Brown (2013). Recently, Sotaniet al. (2014) discussed how for low-mass NSs this could betranslated into a constraint for a particular combination of Kand L. Currently, radius observations are much more modeldependent than mass measurements, largely because radiusmeasurements are much more indirect. Possible sources ofsystematic error include the composition of the atmosphere,the strength of the magnetic field, the distance to the source,interstellar extinction, residual accretion in binaries, bright-ness variations over the surface, and the effects of rotation insources with unknown spin frequencies; see Miller (2013) andPotekhin (2014) for details. The importance of uncertainties indetermining the radius depends on the type of the objectobserved. Currently radii are extracted from four differenttypes of sources as follows:

(1) Isolated neutron stars (INSs). For INSs it is extremelydifficult to determine the distance, the magnetic field,and the composition of the atmosphere inducingaltogether very large uncertainties on the radiusdeterminations from these sources; see, e.g., thediscussion by Potekhin (2014).

(2) Quiescent x-ray transients (QXTs) in low-mass x-raybinaries. The thermal emission from the surface of theNS can be observed in the quiescent phase, i.e., whenthe accretion of matter from the companion is absentor at least strongly reduced. They are promisingsources for radii determinations, since the magneticfield of QXTs is low due to the accretion of matter. Inaddition, the atmosphere is likely to be composed oflight elements (H or possible He) and if they aresituated in globular clusters, the distance is wellknown. Recent radius determinations from QXTsare shown in Fig. 7 as SL13 (Steiner, Lattimer, andBrown, 2013), GS13 and GS13m (Guillot et al.,2013), and GR14 (Guillot and Rutledge, 2014).Although promising sources, the results are stillsubject to many uncertainties. For instance, therehas been recent discussion about the NS’s atmosphericcomposition in quiescent low-mass x-ray binaries(qLMXBs) in the globular cluster NG 6397. Guillotet al. (2013) and Guillot and Rutledge (2014) favoredan unmagnetized hydrogen atmosphere and obtained asmall radius of R1.4 ¼ 9.4� 1.2 km (90% confidencelevel) for a 1.4M⊙ NS (Guillot and Rutledge, 2014).Heinke et al. (2014) argued that a helium atmosphere

is more probable which leads to approximately 2 kmlarger radii. GS13m therefore shows the result ofGuillot et al. (2013) upon excluding the qLMXB inNGC 6397.

(3) Bursting NSs (BNSs). From these objects very power-ful photospheric radius expansion bursts are observed.Similar to QXTs, they have low magnetic fields and alight element atmosphere and, if situated in globularclusters, the distance can be well determined. Themain uncertainties arise here from the modeling of thephotospheric burst and no consensus has yet beenreached; see, e.g., Galloway and Lampe (2012), Özel,Gould, and Güver (2012), Güver and Özel (2013),Steiner, Lattimer, and Brown (2013), and Poutanenet al. (2014). Recent radius determinations from BNSsare shown in Fig. 7; PN14 from Poutanen et al. (2014),GO13 from Güver and Özel (2013), and SL13 fromSteiner, Lattimer, and Brown (2013).

(4) For rotation-powered millisecond pulsars radii can bedetermined from the shape of the x-ray pulses. They areinteresting, in particular, if the mass is known fromradio observations. The result of Verbiest et al. (2008)andBogdanov (2013) for J0437-4715 is shown inFig. 7(B13). Although with large uncertainties, the possiblemass-radius region of neutron star XTE J1807-294 hasbeen derived by Leahy, Morsink, and Chou (2011).

QXTs as well as BNSs are likely to rotate at a frequency of afew hundred of Hz, inducing a non-negligible rotationaldeformation that complicates the analysis of the x-ray spectra.The latter effect is expected to affect the radii by roughly 10%(Poutanen et al., 2014; Bauböck et al., 2015). Özel et al.(2016) included this rotational correction in a combinedanalysis of observed QXTs and BNSs. This common analysisof 12 sources statistically reduces the error on the final resultfor the radius obtained R1.5 ¼ 10.1–11.1 km.In conclusion, present radius determinations are subject to

many assumptions and uncertainties; see also the discussionby Potekhin (2014) and Fortin et al. (2015). Currently, theycannot provide as stringent constraints as some of the massmeasurements. However, much observational efforts are

FIG. 7. Summary of recent NS radius estimations from obser-vations for a star with the canonical mass of 1.4M⊙. Shown are2σ error bars. For details see Table 2 of Fortin et al. (2015) andthe text. From M. Fortin.



directed to NS radius measurements. Future high-precisionx-ray astronomy, such as proposed by the projects NICER,ATHENAþ or LOFT, and the gravitational wave signal of NSmergers expected for the near future (see Sec. VI.A) wouldhelp substantially to constrain radii and consequently the EoSof NS matter.Another interesting possibility to determine a relation

between mass and radius of a NS would be the observationof the gravitational redshift at the NS surface. Cottam, Paerels,and Mendez (2002) deduced a value of z ¼ 0.35 from narrowabsorption lines in the spectra of x-ray bursts from EXO 0748-676. However, this observation could not be confirmed later(Cottam et al., 2008). In addition, the rotation frequency of thesource was measured to be of the order of 400–500 Hz.Therefore, one expects not narrow but wider lines. As aconsequence Lin et al. (2010) concluded that these spectrallines do not actually originate from the surface. However, amore recent study by the same group suggests that line profilesfrom rotating NSs might actually be narrower than initiallypredicted (Bauböck, Psaltis, and Özel, 2013).

2. Neutron star cooling and rotation

While computing mass and radius of a NS requires only aknown relation between total pressure and total energy density[see Eq. (36)], the cooling of NSs depends on a detaileddescription of the interior composition which determines theheat transport and amount of neutrino emission. The occur-rence of superfluid states barely influences the structure of aNS but has great impact on cooling. First, pair breaking andformation are important neutrino emission channels. At a laterstage, for temperatures below the corresponding criticaltemperature, the related pairing gaps suppress the emissionof neutrinos and reduce the heat capacity and thermalconductivity (Blaschke, Grigorian, and Voskresensky, 2004;Yakovlev and Pethick, 2004; Page, Geppert, and Weber,2006). NS cooling and the description of superfluid phaseshave been reviewed by Weber (1999), Yakovlev and Pethick(2004), and Potekhin, Pons, and Page (2015). Althoughcooling calculations face many difficulties due to a largenumber of not precisely known quantities, the direct coolingobservation of the young, only about 330 yr old NS inCassiopeia A (Heinke and Ho, 2010) over a period of 10 yrpromises to give direct insight into its composition (Pageet al., 2011; Shternin et al., 2011; Blaschke et al., 2012;Sedrakian, 2013). A recent analysis of Chandra observationssuggests that the initially reported fast cooling has to beconsidered with caution due to involved statistical uncertain-ties (Posselt et al., 2013) and possible instrumental problems(Elshamouty et al., 2013). Nevertheless, the theoretical workhas demonstrated the strong impact precise cooling observa-tions can have.Neutron star rotation rates can be determined precisely from

pulsar observations. Theoretically, slowly rotating stars can bedescribed in the Hartle and Thorne (1968) approximation; see,e.g., Weber (1999). Numerically precise solutions (Nozawaet al., 1998) can be obtained up to the mass shedding limit, theKepler frequency; see Friedman and Stergioulas (2013). Thevalue of the Kepler frequency depends on the EoS and anobserved frequency above the Kepler limit for a given EoS

would clearly exclude the underlying model. Currentlyobserved rotation rates (Hessels et al., 2006; Kaaret et al.,2007) with a maximum of 716 Hz do not put relevantconstraints on the EoS (Haensel et al., 2009), but this couldchange if more rapidly rotating stars are observed in the future.Other astrophysical observations can be used to derive EoSconstraints, e.g., quasiperiodic oscillations in soft-gamma-rayrepeaters (Steiner and Watts, 2009; Sotani et al., 2012).However, the modeling of these events is complicated andoften relies on additional model assumptions.

D. Summary of constraints on the symmetry energy

Besides the constraints discussed, further constraints on thesymmetry energy at saturation and on the slope parameterhave been collected in the literature in Tsang et al. (2012),Lattimer and Lim (2013), Li and Han (2013), and Lattimerand Steiner (2014). Extending these data collections, thecompilations in Fig. 8 depict the probability distributionsof J and L values, respectively. For simplicity, the probabilitydistributions are assumed to be of Gaussian form with an areanormalized to 1. They are centered at the obtained values for Jand L with widths that are given by the errors of the individual

20 22 24 26 28 30 32 34 36 38 40 42 44symmetry energy at saturation J [MeV]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

prob

abili

ty d

P/dJ

[M

eV-1

] (31.7 +/- 3.2) MeVJ =

0 10 20 30 40 50 60 70 80 90 100 110 120symmetry energy slope coefficient L [MeV]

0.00

0.02

0.04

0.06

0.08

0.10

prob

abili

ty d

P/dL

[M

eV-1

]

L =(58.7 +/- 28.1) MeV

FIG. 8. Probability distribution of the symmetry energies atsaturation J (top panel) and of the symmetry energy slopeparameter L (bottom panel) from various studies. See text fordetails.



studies. Dashed vertical lines are used if no uncertainty isavailable. Allowed ranges with upper or lower bounds areindicated by arrows. In general, nuclear matter parameterssuch as J and L and their errors are correlated in models thatare used in the analysis of experimental and observational

data. Since these correlations are rarely specified in theliterature (Lattimer and Lim, 2013), we treat J and L asindependent quantities. The origin of the constraints isencoded in Fig. 8 by colors; see Table II. Averaging overthis selection of results (excluding upper and lower bounds)we find J ¼ 31.7� 3.2 MeV and L ¼ 58.7� 28.1 MeVwithan error for L that is considerably larger than that for J.

V. MODELING THE EoS

In this section, we describe models for the EoS, i.e.,particular realizations of the formal approaches introducedin Sec. III. It is evident that the requirements on the EoS aredifferent depending on the astrophysical situation to whichthey are applied (see Sec. II.B). We place emphasis on thegeneral purpose EoSs that cover the full thermodynamicparameter range in T, nB, and Yq. An overview of thecurrently available ones is presented in Sec. V.D. The reasonfor this is twofold. First, there is a plethora of different EoSsavailable in the literature applicable in a particular context,especially for cold β-equilibrated NSs. To list all availablemodels, including variations of free parameters such ascoupling constants in the phenomenological models, wouldbe fruitless. Second, excellent reviews already exist; see, e.g.,Lattimer and Prakash (2007), Baldo and Burgio (2012), andLattimer (2012). We therefore discuss only some selectedaspects of EoSs of cold β-equilibrated neutron stars (seeSec. V.A) and of EoSs describing homogeneous matter atfinite temperatures suitable for describing hydrostatic PNSs(see Sec. V.B). Section V.C gives a few representativeexamples of EoSs that describe clusterization and nuclearstatistical ensembles at finite temperatures but that arerestricted to subsaturation densities.Almost exclusively, phenomenological models have been

used up to now in the context of astrophysical applications dueto the computational complexity in the description of clusteredmatter. This concerns the NS crust discussed in Sec. V.A.1, theNSE-type EoSs in Sec. V.C, and, in particular, the generalpurpose EoSs discussed in Sec. V.D. The most advanceddescriptions of densematter are found for particular conditions.In fact, the ab initio approaches discussed in Sec. III, if notrestricted to pure neutron matter, have been applied only tohomogeneous nuclear matter at various neutron-to-protonratios with an interpolation to obtain the β-equilibrated NSEoS. Some ab initio calculations exist at finite temperature assome of the methods, for instance, SCGF, are easier to treat atnonzero temperature, but the composition is fixed andmatter ishomogeneous (see Sec. V.B for some examples). It is desirablethat in the future reliable approaches will be developed todescribe strongly interacting matter for all relevant conditionsneeded in compact star astrophysics. A first step is that theinformation obtained from ab initio neutron matter calcula-tions, experiments, and NS observations is fully exploited toconstrain the general purpose models.

A. Neutron star EoS

The physics of NSs has been discussed in detail in severalworks (Baym and Pethick, 1975, 1979; Glendenning, 1997;Heiselberg and Pandharipande, 2000; Lattimer and Prakash,

TABLE II. Sources of the data for the symmetry energies atsaturation J and slope parameters L used in Fig. 8, including thecolor code.

Type of constraint References

Systematic of nuclearmasses (solid green lines)

Myers and Swiatecki (1996),Danielewicz (2003),Mukhopadhyay and Basu(2007),Klüpfel et al. (2009),Kortelainen et al. (2010),Liu et al. (2010),Möller et al. (2012),Lattimer and Lim (2013),Wang, Ou, and Liu (2013),and Viñas et al. (2014b)

Neutron skin data and othernuclear structureinformation(solid blue lines)

Centelles et al. (2009),Warda et al. (2009),Chen et al. (2010)Chen (2011), Agrawal, De,and Samaddar (2012),Dong et al. (2012), Zhangand Chen (2013), Wang and Li(2013), Danielewicz and Lee(2014), and Viñas et al.(2014b)

Nuclear resonances(solid red lines)

Klimkiewicz et al. (2007),Carbone et al. (2010), Roca-Maza, Brenna et al. (2013),Colo, Garg, and Sagawa(2014), and Paar et al. (2014)

Dipole polarizability of nuclei(solid black lines)

Roca-Maza, Centelles et al.(2013) and Tamii, vonNeumann-Cosel, andPoltoratska (2014)

α and β decay of nuclei(solid orange lines)

Dong et al. (2013) and Dong,Zuo, and Gu (2013)

Global nucleon opticalpotentials (dashed bluelines)

Xu, Li, and Chen (2010)

Heavy-ion collisions(dashed orange lines)

Tsang et al. (2004, 2009), Chen,Ko, and Li (2005a, 2005b), Liand Chen (2005), Shetty,Yennello, and Souliotis (2007),Sun et al. (2010), and Kohleyet al. (2010)

Theoretical calculations(dashed green lines)

Erler, Klüpfel, and Reinhard(2010), Gandolfi, Carlson, andReddy (2012), Fiorilla, Kaiser,and Weise (2012), Erler et al.(2013), Hebeler et al. (2013),Krüger et al. (2013), andNazarewicz et al. (2014)

Properties of neutron stars(dashed red lines)

Newton and Li (2009), Steiner,Lattimer, and Brown (2010,2013), Gearheart et al. (2011),Steiner and Gandolfi (2012),Wen, Newton, and Li (2012),Vidaña (2012), Sotani et al.(2013a, 2013b), and Lattimerand Lim (2013)



2001, 2004, 2007, 2011; Sedrakian, 2007; Chamel andHaensel, 2008; Potekhin, 2010). NS matter is charge neutraland can be considered as cold (T ¼ 0) and in general inβ equilibrium. The EoS thus depends only on one statevariable, which can be conveniently chosen. For example,this can be the baryon number density. The EoS entirelydetermines global properties of stationary NSs, such as massesand radii. For nonrotating stars with negligible magneticfield, they are found by solving the following Tolman-Oppenheimer-Volkoff (TOV) equations (Oppenheimer andVolkoff, 1939; Tolman, 1939):

dPðrÞdr

¼ −G½ϵðrÞ þ PðrÞ�½MðrÞ þ 4πr3PðrÞ�

r2½1 − 2GMðrÞ=r� ;

dMðrÞdr

¼ 4πϵðrÞr2;ð36Þ

relating the gravitational mass of the starM inside a radius r topressure P and energy density ϵ. The EoS in terms of P and ϵcloses the system of equations. Despite the assumptions ofzero temperature and β equilibrium, the calculation of the NSEoS is not a trivial task, particularly if microscopic methodsare applied; see, e.g., Baldo, Bombaci, and Burgio (1997),Vidaña et al. (2010), Schulze and Rijken (2011), and Baldoand Burgio (2012). The domain of validity of some ab initiomethods is restricted to rather low densities not exceedingnuclear saturation density substantially. The size of higher-order contributions in systematic expansions, such as χEFTapproaches, increases with density, and the composition ofmatter at suprasaturation densities is rather uncertain (seeSec. V.A.2). In order to cover the whole density range requiredto describe NSs, microscopic EoSs can be extended at highdensities with generic parametrizations, such as piecewisepolytropes. Thereby the uncertainty of the NS mass-radiusrelation and the dependence on the model parameters can beexplored (Hebeler et al., 2013).

1. Neutron star crust EoSs and unified neutron star EoSs

For mass densities below about 104 g=cm3, an atmosphereof partially ionized atoms and electrons forms the outer part ofa NS with an EoS given by Feynman, Metropolis, and Teller(1949), Rotondo et al. (2011), and de Carvalho et al. (2014).At higher densities, the spatial region that is made up ofinhomogeneous nucleonic matter and electrons not bound tonuclei in β equilibrium is called the crust. It can be dividedinto an outer crust with a plasma of nuclei and electrons asdegrees of freedom and an inner crust where also unboundneutrons exist. Figure 9 gives a graphical representation of thestate of matter in the crust. The results shown employ the EoSof Ruester, Hempel, and Schaffner-Bielich (2006) for theouter crust, where experimentally measured binding energieshave been used in combination with nuclear structure calcu-lations with the SLy4 EDF. For the EoS of the inner crust andthe core, the results of Douchin and Haensel (2001) are taken,which are based on Thomas-Fermi calculations using the sameSLy4 EDF. To obtain the radial structure of the assumed NSwith a mass of 1.44M⊙, the TOVequations (36) were solved.The outer crust is composed of completely ionized nuclei in

a sea of electrons of almost constant density due to the largeincompressibility of the highly degenerate electron fluid. Inthe standard picture, only a single nuclear species exists at agiven density. These nuclei form a bcc lattice of ions asdemonstrated in classical one-component plasma simulations(see Sec. III.C.6). The individual nuclei at their lattice sites canbe identified as the blue dots in Fig. 9. At densities of about107 g=cm3 and below, a crystal of 56Fe nuclei is expected toform. With increasing density the lattice constant decreasesand the electron chemical potential rises substantially. Itbecomes energetically favorable to squeeze electrons intothe nuclei, converting protons to neutrons. A sequence of bcclattices with more and more neutron-rich ions on the latticesites appears the deeper one penetrates into the NS. Eachchange from one to the next nuclear species is connected to a

10.610.710.810.911.011.111.211.311.411.511.6r [km]

outer crust inner crust core

surface neutron drip point crust-core transition

-300

-200

-100

0

100

200

300

z[fm

]0.00.10.20.30.40.50.60.70.80.91.0

nB/n0

FIG. 9. Graphical representation of the structure and composition of the crust of a 1.44M⊙ NS. Each subpanel shows in color coding(see legend at right) the mean local density of the nucleons, for the position in the NS as indicated in the bottom part of the figure. Theillustration uses the SLy4 EDF, the EoS of Ruester, Hempel, and Schaffner-Bielich (2006) for the outer crust, and the EoS of Douchinand Haensel (2001) for the inner crust and the core.



phase transition with a jump in the density (cf. Sec. III.D.2).The series of nuclei in the outer crust is determined by theirmasses and is influenced strongly by shell effects. With therecent progress to measure masses of very neutron-rich nucleiexperimentally with high precision, the order of ions in theouter crust from 56Fe via 62Ni, 64Ni, 66Ni, 86Kr, 84Se, and 82Gecould be established with increasing depth (Kreim et al., 2013;Wolf et al., 2013). For higher densities, the nuclear massesfrom theoretical models, e.g., liquid-drop or EDF type, havebeen used to determine the chemical composition of the outercrust. Since the early works of Salpeter (1961) and Baym,Pethick, and Sutherland (1971) the theoretical description ofthe outer crust is well settled and the main changes result fromimprovements in the theoretical description of exotic nucleinot studied experimentally so far (Haensel, Zdunik, andDobaczewski, 1989; Haensel and Zdunik, 1990; Haenseland Pichon, 1994; Ruester, Hempel, and Schaffner-Bielich,2006; Pearson, Goriely, and Chamel, 2011; Roca-Maza et al.,2012; Wolf et al., 2013).At a mass density of approximately 1011 g=cm3 the neutron

drip density is reached, i.e., the neutron chemical potentialbecomes too high for nuclei at the lattice sites to bindadditional neutrons. In Fig. 9, the contribution of theseunbound neutrons, indicated by the blue color, becomesvisible only at sufficiently high densities. These unboundneutrons can propagate more or less freely through the lattice,although their interaction with the lattice could modify thecrystalline structure (Kobyakov and Pethick, 2014). A propertreatment of the periodic crystal structure and its effect onneutron and electron properties requires a description usingband structure models as in solid state physics (Pethick andThorsson, 1997; Chamel, 2005). Since the temperature is verylow, effects of neutron pairing could be important. This is lessrelevant for the basic thermodynamic properties of the crustmatter itself but it has to be considered for dynamic processesand thermal properties, in particular, neutron star cooling. Inview of neutron superfluidity, the so-called entrainment effecthas to be taken into account in hydrodynamic descriptions ofthe NS’s inner crust and core. In this case the momentum ofone fluid is not aligned with its particle current, but dependson the particle currents of all other fluids (Carter, Chamel, andHaensel, 2005a, 2005b, 2006; Chamel, 2005; Gusakov andHaensel, 2005; Chamel and Haensel, 2006; Gusakov, Kantor,and Haensel, 2009b).The exact location of the neutron drip density is sensitive to

details of the theoretical model, in particular, to the isospindependence of the effective interaction due to the very largeneutron excess encountered in the crust; see, e.g., Douchin andHaensel (2000, 2001), Steiner (2008), and Ducoin et al.(2011). The properties of nuclei also change inside matter athigh densities, mostly in the inner crust of the NS when theyare surrounded by a gas of neutrons and the electric field isscreened by the electrons.Several studies have been devoted to the description

of nuclei in a dense medium and the effects on the EoS;see, e.g., Baym, Bethe, and Pethick (1971), Barkat, Buchler,and Ingber (1972), Ravenhall, Bennett, and Pethick (1972),Negele and Vautherin (1973), Lamb et al. (1978), Cheng, Yao,and Dai (1997), Baiko and Haensel (1999), Douchin and

Haensel (2001), Matsuzaki (2006), Ducoin et al. (2008),Papakonstantinou et al. (2013), Aymard, Gulminelli, andMargueron (2014), and Raduta, Aymard, and Gulminelli(2014). In most cases, the Wigner-Seitz approximation inspherical cells surrounding a single nucleus is employed (seeSec. III.C.6). With increasing depth inside the crust, nucleiapproach each other and the action of the short-range nuclearinteraction beyond the size of an individual nucleus has to beconsidered. First, a strong deformation of nuclei and a changein the shell structure is observed in model calculations(Oyamatsu, 1993, 1994; Douchin, Haensel, and Meyer,2000), such that they finally touch and the sequence ofclassical pasta phases is found (Buchler and Barkat, 1971;Ravenhall, Pethick, andWilson, 1983; Watanabe et al., 2003b,2009; Maruyama et al., 2005; Avancini et al., 2009; Newtonand Stone, 2009). The picture of the classical pasta phaseswith their specific geometries and phase transitions changes ifmore general shapes are allowed in full three-dimensionalcalculations with less restrictions on the symmetries(Watanabe et al., 2005; Nakazato, Oyamatsu, and Yamada,2009; Okamoto et al., 2012; Schneider et al., 2014;Schuetrumpf et al., 2015). The extension of the inner crust,the types of pasta phases, and the transition density to uniformmatter depends crucially on the density dependence of thesymmetry energy (Pethick, Ravenhall, and Lorenz, 1995;Oyamatsu and Iida, 2007; Roca-Maza and Piekarewicz, 2008;Kubis, Porebska, and Alvarez-Castillo, 2010; Grill,Providência, and Avancini, 2012; Grill et al., 2014). Thepasta phase could be relevant for the neutrino transport in thePNS and the subsequent cooling of the NS. For example,Horowitz et al. (2015) showed that the pasta phase can reducethe electrical and the thermal conductivities. The reducedelectrical conductivity might be related to the observed upperlimit of x-ray pulsar spin periods (Pons, Viganò, andRea, 2013).The crust has a subdominant effect on global properties of

NSs such as mass or radius. Therefore, a crust EoS is matchedoften to an EoS of uniform matter from an independent modelcalculation. Considerable effort is required in developing anEoS which describes matter from the surface to the center ofthe NS in a unified manner, i.e., on the basis of the sameinteraction model, including a description of inhomogeneousmatter in the crust. This is important for detailed predictionsof NS radii and for dynamical properties. Only few suchunified NS EoSs exist; see, e.g., Douchin and Haensel (2001),Fantina et al. (2013), Miyatsu, Yamamuro, and Nakazato(2013), Baldo et al. (2014), Gulminelli and Raduta (2015),and Sharma et al. (2015). Unified NS EoSs can also beobtained from the general purpose EoS discussed in Sec. V.Dby applying zero (or negligibly small) temperature andβ-equilibrium conditions. However, the aforementioneddedicated unified NS EoS models often give a more detaileddescription of nonuniform NS matter. A comparison of theseclasses of models is useful to investigate limitations of generalpurpose EoSs regarding their description of nuclei in denseand cold matter.In contrast to a conventional star in a hadronic model, the

EoS is very different for strange stars (Alcock, Farhi, andOlinto, 1986; Haensel, Zdunik, and Schaeffer, 1986), and thestructure of the crust is still a matter of debate (Alford et al.,



2006; Jaikumar, Reddy, and Steiner, 2006; Oertel andUrban, 2008).

2. Composition of the neutron star core

The composition of matter at suprasaturation densitiesreached in the NS core is uncertain and, in particular, particlesother than nucleons and electrons are expected to appear. Inthe literature muons, pions, kaons and their condensates,hyperons, nuclear resonances, and quarks have been consid-ered (Glendenning, 1997). There is even the possibility ofabsolutely stable strange quark matter (Farhi and Jaffe, 1984;Witten, 1984) and pure strange stars (Alcock, Farhi, andOlinto, 1986; Haensel, Zdunik, and Schaeffer, 1986); see alsoItoh (1970). In this context, the recent discovery of two NSswith masses of about 2M⊙ (Demorest et al., 2010; Antoniadiset al., 2013; Fonseca et al., 2016) has triggered intensivediscussions, since without an interaction, any additionaldegree of freedom softens the EoS simply by lowering theFermi energies of the particles present. As a consequence, alower maximum mass is obtained and many older modelscontaining additional particles are in contradiction with theNS mass constraint.Phenomenological quark models can easily be supple-

mented with the necessary repulsion at high densities. Asan example, for the NJL model Lagrangian of Eq. (27) this canbe achieved by adding a vector interaction term of the form

LV ¼ GVðψγμψÞðψγμψÞ: ð37Þ

Maximum NS masses above 2M⊙ can then be obtained; see,e.g., Alford et al. (2007), Klähn et al. (2007), Weissenbornet al. (2011), Zdunik and Haensel (2013), and Buballa et al.(2014). Masuda, Hatsuda, and Takatsuka (2013) proposed thatthe transition from hadronic to quark matter might be acrossover potentially leading to an increase of the maximummass. However, this scenario requires an ad hoc interpolationscheme to connect the hadronic and the quark phase (Kojoet al., 2015). The same scenario was recently applied in thecontext of PNSs by Masuda, Hatsuda, and Takatsuka (2016).Hyperonic degrees of freedom are more difficult to recon-

cile with a 2M⊙ NS. Most models that include hyperonspredict that they appear at nB ∼ ð2 − 3Þnsat but lead at thesame time to maximum NS masses of ∼1.4M⊙, well belowthe highest observed ones. Sometimes this is called the“hyperon puzzle” in the literature (Lonardoni et al., 2015).A similar effect is observed with nuclear resonances (Dragoet al., 2014) and meson condensates. It is thus obvious thatadditional repulsion is needed to stiffen the high-density EoS.Different solutions have been proposed to overcome this

problem. The first one is that a transition to quark matterappears at sufficiently low densities such that hyperons orother additional hadronic particles have not yet softened theEoS too much. A two family scenario with low mass compacthadronic stars and high mass quark stars was recentlydiscussed by Drago, Lavagno, and Pagliara (2014).Another possibility is to modify the interactions at high

densities. Hyperonic interactions have been extensively stud-ied in this respect. Since experimental data are scarce andfurnish only weak constraints on the interactions at

subsaturation densities, even less is known about thehyperon-nucleon (YN) and hyperon-hyperon (YY) inter-actions at the relevant densities in the core of NSs (seeSec. III.A.1). Presently several phenomenological EoS modelsexist that contain hyperons and predict maximum NS massesin agreement with observations; see, e.g., Hofmann, Keil, andLenske (2001a), Rikovska-Stone et al. (2007), Bednarek et al.(2012), Bonanno and Sedrakian (2012), Weissenborn,Chatterjee, and Schaffner-Bielich (2012a, 2012b), Colucciand Sedrakian (2013), Banik, Hempel, and Bandyopadhyay(2014), Lopes and Menezes (2014), van Dalen, Colucci, andSedrakian (2014), Gomes et al. (2015), and Oertel et al.(2015). The crucial point is that the interaction is adjusted toprovide the necessary repulsion.In microscopic models the missing repulsion for hyperons

is more difficult to obtain. Naturally, one would expect it toarise from three-body forces. But, using a microscopic modelbased on the BHF approach, Vidaña et al. (2011) found thateven adding a phenomenological three-body force was notenough to allow for the existence of stars that are massiveenough to be compatible with observations. Recent relativ-istic DBHF calculations (Katayama and Saito, 2014), includ-ing automatically part of the three-body forces, reproducehyperonic NSs with two solar masses, but with a nuclear EoSthat is either too stiff or does not give enough binding incontradiction with known properties of symmetric nuclearmatter at saturation. On the other hand, in recent calculationsusing an auxiliary field diffusion Monte Carlo method(AFQMC) (Lonardoni, Gandolfi, and Pederiva, 2013;Lonardoni, Pederiva, and Gandolfi, 2014; Lonardoni et al.,2015), it was found that a sufficiently strong repulsive three-body force, constrained by the systematics of separationenergies in a series of hypernuclei, can produce an EoS stiffenough to satisfy the 2M⊙ constraint, even if a strong modeldependence due to the phenomenological nature of thehyperonic two- and three-body forces is apparent. In con-clusion, there are still many open questions regarding the roleof hyperons and other additional non-nucleonic degrees offreedom in NSs.

B. EoS of uniform matter at finite temperature

The thermal properties of nuclear matter are an importantsubject on their own and many studies are actually performedwithout an astrophysical application. Examples are the ab ini-tio calculations of neutron matter at finite temperatures, someof which have been mentioned in Sec. IV.B, or studies ofthe nuclear liquid-gas phase transition which occurs atsubsaturation densities if Coulomb and finite-size effectsare neglected. The liquid-gas phase transition is an importantaspect of the low-density nuclear matter EoS and has beenstudied extensively in the literature; see, e.g., Barranco andBuchler (1980) and Müller and Serot (1995). An example isshown in Fig. 4. In addition to BHF (Baldo and Ferreira, 1999;Baldo, Ferreira, and Nicotra, 2004) and DBHF (Ter Haar andMalfliet, 1987; Huber, Weber, and Weigel, 1998) calculationsextended to finite temperature, consistent SCGF calculationshave been reported by Rios et al. (2008) and Fiorilla, Kaiser,and Weise (2012). Wellenhofer et al. (2014) and Wellenhofer,



Holt, and Kaiser (2015) studied the phase diagram of nuclearmatter applying in-medium χEFT.Here we are mainly interested in EoSs relevant for astro-

physical applications. Finite-temperature effects are of par-ticular relevance for PNSs, CCSNe, and NS mergers and canbe studied in HICs too. Although more scarce than NS EoSs,there is a variety of works considering EoSs of homogeneousmatter at fixed entropies or temperatures and hadronic chargefractions. Many of these EoSs have actually been developedfor studying PNSs by considering characteristic hydrostaticconfigurations that represent different evolutionary stages.The history concerning additional particles, such as hyperons,mesons, or quarks, is almost as long as for cold neutron stars,reaching from the discussion of quark matter formation ormeson condensates to hyperons; see Prakash et al. (1997) foran early review and Pons et al. (2000), Pons, Miralles et al.(2001), Bombaci et al. (2007), Menezes and Providência(2007), Dexheimer and Schramm (2008), Yasutake andKashiwa (2009), Beisitzer, Stiele, and Schaffner-Bielich(2014), and Masuda, Hatsuda, and Takatsuka (2016) for afew examples.Often PNSs are approximated as isentropic, i.e., having a

constant entropy per baryon s, and/or a fixed lepton orelectron fraction (YLðeÞ or Ye, respectively) with or withouttrapped neutrinos. Sometimes isothermal PNS are consideredtoo. If the crust and envelope of the PNS are neglected, thesituation is still similar to that for cold NS: Matter is uniformand for each parameter combination of s or T and YLðeÞ or Ye

the EoS is still one dimensional, i.e., it depends only on onestate variable, e.g., baryon number density.Most ab initio calculations of the EoS of nuclear matter at

finite temperature concern pure neutron matter (see Sec. IV.B),or symmetric nuclear matter, since general asymmetric nuclearmatter asks for more involved computations. Some exceptionsexist, e.g., the EoS of Togashi and Takano (2013) uses thevariational method, starting from a nuclear Hamiltonian that iscomposed of the Argonne v18 and Urbana IX potentials. Theyaim at providing a full general purpose EoS that can be appliedin astrophysical simulations (see Sec. V.D). However, in theirfirst work, they considered only uniform nuclear matter, butfor various temperatures and asymmetries. To simplify com-putations, the frozen-correlation approximation is employed:the self-energies and correlation matrix elements are evaluatedat zero temperature. This approximation is motivated by theresults of Baldo and Ferreira (1999). Togashi and Takano(2013) validated it by comparing with results for fullyminimized calculations.The frozen-correlation approximation is standard in finite-

temperature BHF calculations aimed to model the PNS EoStoo; see, e.g., Nicotra et al. (2006a, 2006b), Burgio, Schulze,and Li (2011), and Chen et al. (2012). Effects of hyperonsand/or quarks were considered too within these works relyingon different models for the interactions. For instance, for thequark phase Chen et al. (2012) used a model based on theDyson-Schwinger equations of QCD (cf. Sec. III.B.1.h),whereas Nicotra et al. (2006b) applied the MIT bag model.As known for BHF calculations (see Sec. V.A.2), themaximum mass of a cold NS including additional degreesof freedom such as hyperons or quarks lies in general well

below the canonical value of 2M⊙. Exceptions are the hybridNS and PNS models of Chen et al. (2012) where much highermaximum masses in the vicinity of 2M⊙ were found.Applying phenomenological interactions to finite-

temperature matter, an obvious question is whether theeffective couplings, determined via zero temperature proper-ties of strongly interacting (mainly nuclear) systems, dependon temperature. This was addressed by Fedoseew and Lenske(2015), where the thermal properties of asymmetric nuclearmatter have been analyzed within a relativistic approach. Theparameters of a density-dependent relativistic hadron nuclearfield theory, similar to a density-dependent RMF model, havebeen adjusted to reproduce DBHF results for in-mediumself-energies. In particular, it was shown that the temperaturemodifications of the nucleon-meson couplings is almostnegligible. A comparison of the free energy of nuclear matterusing the Brussels Skyrme interaction with the results ofFiorilla, Kaiser, and Weise (2012) leads to the same con-clusion (Fantina, 2015). In Moustakidis and Panos (2009),where a momentum-dependent finite-range term is added to aSkyrme-type interaction, temperature-dependent couplingsare obtained but this dependence is weak up to temperaturesof 30 MeV.Constantinou et al. (2014) thoroughly investigated the

finite-temperature properties of the bulk EoS. They employedthe potential model of Akmal, Pandharipande, and Ravenhall(1998), which is fitted to results from variational calculations,and compared it with the typical Skyrme EDF SKa fromKöhler (1976). The latter parametrization is also applied inthe H&W EoS (see Sec. V.D.1.a). Analytical formulas arederived for all thermodynamic state variables and theirderivatives at finite temperatures, simplifying the use inastrophysical applications. A similar study of the thermalproperties of the EoS but for finite-range interactions wasrecently published by Constantinou et al. (2015).

C. EoS of clustered matter at finite temperatures

Complementary to investigations of bulk properties ofwarm and dense uniform matter, there are many worksstudying inhomogeneous warm matter within NSE basedmodels (see Sec. III.C.1). Most of them do not cover thefull parameter space relevant for simulations of CCSNe orNS mergers, partly since they are designed for a particularapplication, e.g., multifragmentation experiments or nucleo-synthesis aspects. A typical problem is the omission ofinteractions and/or medium modifications of nuclei. As aresult, the EoS does not provide a realistic description at highdensities. Nevertheless, these models allow one to investigateimportant aspects of the EoS of clustered matter, e.g., thechemical composition and the role of excited states.The impact of the different model ingredients depends onthe thermodynamic conditions.The chemical composition of matter at baryon densities

roughly below 10−3 fm−3 and a few MeV temperature ismainly driven by the nuclear binding energies and the treat-ment of thermal excitations. Simple mass formulas providebinding energies for the widest possible range of nuclei thatare considered in statistical models. A liquid-drop-type massformula is used in the statistical model for supernova matter



(SMSM) (Botvina and Mishustin, 2004, 2010; Buyukcizmeci,Botvina, and Mishustin, 2014). It is based on the SMM(Bondorf et al., 1995; Sagun et al., 2014) which has provensuccessful in the analysis of fragment yields in low-energyHICs. The parameters of the used liquid-drop mass formula,including temperature effects, have been calibrated by theanalysis of experimental multifragmentation data. A liquid-drop parametrization is also used in the statistical model ofRaduta and Gulminelli (2009, 2010). The NSE model ofBlinnikov et al. (2011) considered up to 20 000 nuclei, whosebinding energies are taken from the theoretical mass formulaof Koura et al. (2005). Binding energies from Myers andSwiatecki (1990, 1994) are adopted in the work of Ishizuka,Ohnishi, and Sumiyoshi (2003) incorporating about 9000different nuclei. Results of the microscopic-macroscopicfinite-range droplet model (Möller et al., 1995) or theDuflo-Zuker model (Duflo and Zuker, 1995) have been usedin Gulminelli and Raduta (2015). Tables with theoreticalmasses from fully microscopic models, mainly EDFs, usuallycover a smaller range of nuclei. Experimental binding energies(Audi, Wapstra, and Thibault, 2003; Wang et al., 2012) areavailable for an even smaller number of nuclei rather close tothe valley of stability. They are used as far as available in someNSE based models, but have to be supplemented by theoreticalmasses for more exotic nuclei. The choice of different sourcescan result in artificial jumps of the isotopic abundances at theboundaries; see, e.g., Buyukcizmeci et al. (2013).As an illustration, we compare in Fig. 10 nuclear abun-

dances for typical conditions in the collapse phase of a CCSNfor three statistical models. The different predictions mainlyreflect the extrapolation of binding energies to neutron-richnuclei. This depends on the choice of the mass model since themost abundant nuclei are situated outside the region wheremasses are experimentally known. A monomodal or bimodalstructure is obtained, peaked around magic numbers inFigs. 10(b) and 10(c), demonstrating the importance of shelleffects, which are missing in models based on simple massformulas (Raduta and Gulminelli, 2010; Buyukcizmeci,Botvina, and Mishustin, 2014), as in Fig. 10(a) or modelsthat use the Thomas-Fermi approximation for the description ofnuclei, as, e.g., in the model of Aymard, Gulminelli, andMargueron (2014). Broad and even bimodal distributionscannot be represented within the SNA (see Sec. III.C.2), whichis employed, e.g., in the general purpose models STOS orLS220, discussed in Sec. V.D. Nevertheless, the predictions forthe average heavy nucleus show only a moderate deviationcompared to that of the statistical HS(DD2) model and globalthermodynamic quantities are only slightly modified (Burrowsand Lattimer, 1984). However, these differences in thecomposition are relevant for electron-capture reactions duringthe collapse phase and thus can influence the dynamicsof a CCSN as discussed in Sec. VI.B.1. For very neutron-rich conditions, the range of nuclei considered in the table,indicated by the gray regions in Fig. 10, also matters. Differentrules for determining the boundary are employed, e.g., vanish-ing neutron separation energies or binding energies.With increasing temperature, excited states of nuclei are

populated. These are considered explicitly in models thatuse temperature-dependent degeneracy factors. Usually, leveldensities of a Fermi-gas type are employed, e.g., those of

0

10

20

30

40

50

60

70

Z

0 10 20 30 40 50 60 70 80 90 100 110 120

N

log10(XN,Z)

-10-8-6-4-20

(a) SMSMnB = 5.69 10-4 fm-3

T = 1.60 MeVYe = 0.35

0

10

20

30

40

50

60

70

Z

0 10 20 30 40 50 60 70 80 90 100 110 120

N

log10(XN,Z)

-10-8-6-4-20

(b) Gulminelli and Raduta (2015)

0

10

20

30

40

50

60

70

Z

0 10 20 30 40 50 60 70 80 90 100 110 120

N

HS(DD2)STOSLS220

log10(XN,Z)

-10-8-6-4-20

(c) HS(DD2)

FIG. 10. Composition of matter in the center of a CCSN 6 msbefore bounce at thermodynamic conditions taken from asimulation of Perego et al. (2015). The color map showsthe distribution of nuclei (mass fractions) in the (a) SMSM(Buyukcizmeci, Botvina, and Mishustin, 2014), the (b) EoS ofGulminelli and Raduta (2015), and the (c) HS(DD2) model.(c) The black cross indicates the average heavy nucleus. Theblack diamond and triangle show the representative heavynucleus of STOS and LS220, respectively, calculated withinthe SNA.



Iljinov et al. (1992) in Raduta and Gulminelli (2009, 2010) orFái and Randrup (1982) in Ishizuka, Ohnishi, and Sumiyoshi(2003) (see Sec. III.C.1.b). An alternative approach is toincorporate temperature effects directly in the mass model,e.g., in the SMSM, by introducing a temperature dependencein the coefficients of the mass formula, in particular, in bulkand surface contributions to the energy. Part of the differencesin Fig. 10 also result from the treatment of excited states. Theactual treatment in the model also influences the dissolutionof heavy clusters with increasing temperature and the changeof the abundance distributions. They are dominated morestrongly by light clusters at high T, which are usually handledindependently of the heavier nuclei using experimental bind-ing energies and sometimes correct quantum statistics. At verylow densities, the finite-temperature EoS is given modelindependently by the VEoS (see Sec. III.C.3). The VEoSwith light species (p, n, d, t, 3He, and α) was discussed byPratt, Siemens, and Usmani (1987) for conditions in HICs.Horowitz and Schwenk (2006a) considered neutrons, protons,and α particles as basic constituents and experimental infor-mation on binding energies and phase shifts was used in thecalculation of the second virial coefficients. O’Connor et al.(2007) added 3H and 3He nuclei, and even heavier specieswere included by Mallik et al. (2008). A relativistic VEoS wasalready given by Venugopalan and Prakash (1992) for aninteracting gas of nucleons, pions, and kaons. A generalfinding, present in all models, is that deuterons, tritons, andhelions appear abundantly for typical conditions of supernovamatter in addition to α particles (Sumiyoshi and Röpke, 2008;Heckel, Schneider, and Sedrakian, 2009; Typel et al., 2010;Hempel et al., 2015; Pais, Chiacchiera, and Providência,2015). The presence of light clusters can modify weakinteraction rates and therefore the dynamics of astrophysicalprocesses (see Sec. VI).With increasing baryon density, above approximately

10−3 fm−3, interaction effects start to play a role and mediumeffects on the binding energies have to be incorporated in thestatistical models. They modify the chemical composition incomparison to results of pure NSE models that use vacuumbinding energies. In essentially all models, the screening ofthe Coulomb potential due to the electron component istaken into account in the Wigner-Seitz approximation (seeSec. III.D.2). The repulsion of nuclei is often modeled by anexcluded-volume mechanism (see Sec. III.C.1.e), e.g., inRaduta and Gulminelli (2009) and Sagun et al. (2014).Raduta and Gulminelli (2010) solved the cluster partitionsums by using Metropolis Monte Carlo techniques whichallow one to consider configuration dependent excluded-volume corrections. A comparison of predictions from thegeometric excluded-volume approach with those of the moremicroscopically inspired approach using mass shifts (seeSec. III.C.4) was presented by Hempel et al. (2011) concen-trating on light nuclei. The dissolution of clusters withincreasing density cannot be described properly in basicstatistical models unless an excluded-volume mechanism ormass shifts are considered. The treatment of the homogeneousnucleonic matter contribution is relatively similar in mostNSE based EoS models, employing, if at all, phenomeno-logical mean-field approaches with various interactions; see

Sec. V.D.3 for a discussion of the compatibility with presentday constraints. Interactions of unbound nucleons are notincluded in the SMSM, therefore it is applicable only to dilutematter. A comparison of different NSE-type models can befound in Ishizuka, Ohnishi, and Sumiyoshi (2003) andBuyukcizmeci et al. (2013) with detailed and comprehensiveanalyses of the nuclear composition.

D. General purpose equations of state

In this section we describe EoSs which cover the fullthermodynamic parameter range necessary for astrophysicalsimulations of CCSNe or NS mergers. Such EoSs not onlyhave to be available for finite temperatures and differentcharge fractions, but should include a description of nonuni-form matter at subsaturation densities, where nuclei appear,and a description of homogeneous matter at high densitiesand/or temperatures.There exist only a few such EoSs. To give an overview, we

summarize their particle content, disregarding leptonicdegrees of freedom here and in the following, and somekey properties for cold NSs of the different models inTable III. We indicate if the EoS is publicly available intabulated form or as a computer code (see Appendix A.1 for alist of different online resources). Key nuclear matter proper-ties of the nuclear interaction models, which are used in theEoS models of Table III, are given in Table IV.We remark that all of the presently available general

purpose EoSs are included in the discussion, even thoughmany of them are in strong disagreement with some astro-physical, experimental, or theoretical constraints. However,several of them are still used for reference applications. Any“benchmarking” of EoSs depends on which constraints arechosen from the many available in the literature, and there isnot a single model that fulfills all of them, not even the verylimited set of constraints that we consider later. A generalpurpose EoS with particular deficits can be interesting becauseit very much depends on the astrophysical context and thespecific application whether a constraint is relevant or not. Forexample, cluster formation at low densities seems to be moreimportant than the neutron matter EoS for the dynamics ofCCSNe (see Sec. VI.B.1) but in NS mergers probably theopposite is the case. Because of the limited number of generalpurpose EoSs and their importance for astrophysical appli-cations, we first give a complete overview. A critical dis-cussion follows at the end in Sec. V.D.3.

1. Nucleons and nuclei as degrees of freedom

a. H&W

The EoS of Hillebrandt, Nomoto, and Wolff (1984) andHillebrandt and Wolff (1985) (H&W) is one of the first EoSsthat was suitable for CCSNe simulations and which is still inuse today (Janka, 2012a). At low densities, a NSE modelbased on the work of El Eid and Hillebrandt (1980) is appliedincluding 470 different nuclei: in addition to neutrons,protons, and α particles, about 450 isotopes with chargenumbers Z between 10 and 32 and neutron numbersN rangingfrom stability to neutron drip as well as 20 heavier nuclei fromthe Zr and Pb region are included. To account for excited



TABLE III. Characteristic properties of the currently existing general purpose EoSs. Top part: EoSs containing nucleons and nuclei; bottompart: EoSs including additional hadronic or quark degrees of freedom. Listed are the nuclear interaction model used, the included particledegrees of freedom, the maximum mass Mmax of cold, spherical (nonrotating) NSs, and their radii at a fiducial gravitational massMG of 1.4M⊙. We included in addition the compactness Ξ ¼ GMG=R of the maximum mass configuration. Empty entries indicate that thevalue was not available to the authors. In nuclear interactions labeled with *, the nucleon masses have been changed to experimental valueswithout a refitting of the coupling constants. This induces a marginal change of the interaction.

ModelNuclear

interactionDegrees offreedom

Mmax,(M⊙)

R1.4M⊙ ,(km) Ξ

Publiclyavailable References

H&W SKa n;p;α;fðAi;ZiÞg 2.21a 13.9a No El Eid and Hillebrandt (1980)and Hillebrandt, Nomoto,and Wolff (1984)

LS180 LS180 n; p; α; ðA; ZÞ 1.84 12.2 0.27 Yes Lattimer and Swesty (1991)LS220 LS220 n; p; α; ðA; ZÞ 2.06 12.7 0.28 Yes Lattimer and Swesty (1991)LS375 LS375 n; p; α; ðA; ZÞ 2.72 14.5 0.32 Yes Lattimer and Swesty (1991)STOS TM1 n; p; α; ðA; ZÞ 2.23 14.5 0.26 Yes Shen et al. (1998a, 1998b,

2011)FYSS TM1 n;p;d;t;h;α;fðAi;ZiÞg 2.22 14.4 0.26 No Furusawa, Sumiyoshi et al.

(2013)HS(TM1) TM1* n;p;d;t;h;α;fðAi;ZiÞg 2.21 14.5 0.26 Yes Hempel and Schaffner-Bielich

(2010) and Hempel et al.(2012)

HS(TMA) TMA* n;p;d;t;h;α;fðAi;ZiÞg 2.02 13.9 0.25 Yes Hempel and Schaffner-Bielich(2010)

HS(FSU) FSUgold* n;p;d;t;h;α;fðAi;ZiÞg 1.74 12.6 0.23 Yes Hempel and Schaffner-Bielich(2010) and Hempel et al.(2012)

HS(NL3) NL3* n;p;d;t;h;α;fðAi;ZiÞg 2.79 14.8 0.31 Yes Hempel and Schaffner-Bielich(2010) and Fischer, Hempelet al. (2014)

HS(DD2) DD2 n;p;d;t;h;α;fðAi;ZiÞg 2.42 13.2 0.30 Yes Hempel and Schaffner-Bielich(2010) and Fischer, Hempelet al. (2014)

HS(IUFSU) IUFSU* n;p;d;t;h;α;fðAi;ZiÞg 1.95 12.7 0.25 Yes Hempel and Schaffner-Bielich(2010) and Fischer, Hempelet al. (2014)

SFHo SFHo n;p;d;t;h;α;fðAi;ZiÞg 2.06 11.9 0.30 Yes Steiner, Hempel, and Fischer(2013)

SFHx SFHx n;p;d;t;h;α;fðAi;ZiÞg 2.13 12.0 0.29 Yes Steiner, Hempel, and Fischer(2013)

SHT(NL3) NL3 n;p;α;fðAi;ZiÞg 2.78 14.9 0.31 Yes Shen, Horowitz, and Teige(2011)

SHO(FSU) FSUgold n;p;α;fðAi;ZiÞg 1.75 12.8 0.23 Yes Shen, Horowitz, andO’Connor (2011)

SHO(FSU2.1) FSUgold2.1 n;p;α;fðAi;ZiÞg 2.12 13.6 0.26 Yes Shen, Horowitz, andO’Connor (2011)

LS220Λ LS220 n;p;α;ðA;ZÞ;Λ 1.91 12.4 0.29 Yes Oertel, Fantina, and Novak(2012) and Gulminelli et al.(2013)

LS220π LS220 n;p;α;ðA;ZÞ;π 1.95 12.2 0.29 No Oertel, Fantina, and Novak(2012) and Peres, Oertel,and Novak (2013)

BHBΛ DD2 n;p;d;t;h;α;fðAi;ZiÞg;Λ 1.96 13.2 0.25 Yes Banik, Hempel, andBandyopadhyay (2014)

BHBΛϕ DD2 n;p;d;t;h;α;fðAi;ZiÞg;Λ 2.11 13.2 0.27 Yes Banik, Hempel, andBandyopadhyay (2014)

STOSΛ TM1 n; p; α; ðA; ZÞ;Λ 1.90 14.4 0.23 Yes Shen et al. (2011)STOSYA30 TM1 n; p; α; ðA; ZÞ; Y 1.59 14.6 0.17 Yes Ishizuka et al. (2008)STOSYA30π TM1 n;p;α;ðA;ZÞ;Y;π 1.62 13.7 0.19 Yes Ishizuka et al. (2008)STOSY0 TM1 n; p; α; ðA; ZÞ; Y 1.64 14.6 0.18 Yes Ishizuka et al. (2008)STOSY0π TM1 n;p;α;ðA;ZÞ;Y;π 1.67 13.7 0.19 Yes Ishizuka et al. (2008)STOSY30 TM1 n; p; α; ðA; ZÞ; Y 1.65 14.6 0.18 Yes Ishizuka et al. (2008)STOSY30π TM1 n;p;α;ðA;ZÞ;Y;π 1.67 13.7 0.19 Yes Ishizuka et al. (2008)STOSY90 TM1 n; p; α; ðA; ZÞ; Y 1.65 14.6 0.18 Yes Ishizuka et al. (2008)STOSY90π TM1 n;p;α;ðA;ZÞ;Y;π 1.67 13.7 0.19 Yes Ishizuka et al. (2008)STOSπ TM1 n; p; α; ðA; ZÞ; π 2.06 13.6 0.26 No Nakazato, Sumiyoshi, and

Yamada (2008)

(Table continued)



states, nuclear level densities have been constructed based onHF potentials using the grand partition function approach(Huizenga and Moretto, 1972; Wolff, 1980). For densitiesabove 3 × 1012 g=cm3, the EoS is computed in the SNA (seeSec. III.C.2), using the thermal HF method. The nuclearinteraction is of the Skyrme type using the parameter set SKa(Köhler, 1976).

b. LS

The EoS by Lattimer and Swesty (1991) (LS) considersnucleons, α particles, and heavy nuclei in the SNA (seeSec. III.C.2) as degrees of freedom. The latter are describedwith a medium-dependent liquid-drop model. For nucleons,nonrelativistic Fermi-Dirac statistics is used, and a simpli-fied momentum-independent nucleon-nucleon interaction isemployed which results in constant effective nucleon massesequal to the applied vacuum masses. Interactions between thegas of nucleons, α particles, and heavy nuclei are taken intoaccount through an excluded-volume mechanism. α particlesare treated as hard spheres of volume vα ¼ 24 fm3 forming anideal Boltzmann gas, neglecting excited states. As the densityincreases, nuclei undergo geometrical shape deformations,until they dissolve in favor of homogeneous nuclear matterabove approximately saturation density. The formation ofnonspherical nuclei and bubble phases is described bymodifying the Coulomb and surface energies of nuclei. Thephase transition to bulk nuclear matter is treated by a Maxwellconstruction between the two phases.The LS EoS exists for three different parametrizations of the

nucleonic interaction, which are usually denoted according totheir value of the incompressibilityK of 180, 220, and375MeV.Nowadays the version with K ¼ 220 MeV is considered themost relevant of the three, since it is the best compatiblewith thevarious constraints on the EoS (see Sec. V.D.3).

c. STOS

The EoS by Shen et al. (1998a, 1998b, 2011) (STOS) isanother widely used general purpose EoS. It assumes the samedegrees of freedom as the LS EoS: neutrons, protons, and

α particles as well as one heavy nucleus in the SNA. Fornucleons a RMFmodel with nonlinear meson self-interactionsis used with the parametrization TM1 (Sugahara and Toki,1994). α particles are again described as an ideal Maxwell-Boltzmann gas with excluded-volume corrections. Excitedstates of α particles are neglected. The properties of therepresentative heavy nucleus are obtained from Wigner-Seitzcell calculations within the Thomas-Fermi approximation forparametrized density distributions of nucleons and α particles.The translational energy and entropy contribution of heavynuclei is not taken into account.Zhang and Shen (2014) investigated the accuracy of the

parametrized density distributions of STOS in comparisonwith fully self-consistent Thomas-Fermi calculations. Theyconcluded that overall there are only small differences. Indetail it was found that the free energies of the original STOSEoS are slightly too low compared to the self-consistentsolutions. This and other differences were related to a toosmall value of the coefficient of the (surface) gradient energyof the density distribution used in STOS for the description ofnuclei, which is not consistent with the employed TM1interaction. In addition, Zhang and Shen (2014) studied theeffect of a possible bubble phase for the transition to uniformnuclear matter and found that the transition can be shifted toslightly higher densities.

d. FYSS

The EoS of Furusawa et al. (2011) and Furusawa, Sumiyoshiet al. (2013) (FYSS) can be seen as an extension of the STOSEoS at subsaturation densities. The same RMF parametrizationTM1 is employed for the nuclear interaction as in the STOSEoS. A distribution of various light nuclei and heavy nuclei upto Z ∼ 1000 is included. Heavy nuclei are not described by theThomas-Fermi approximation as in STOSbut by a liquid-drop-type formulation with temperature-dependent bulk energies.Shell effects are incorporated by extracting the difference of theliquid-drop binding energies compared to experimental (Audi,Wapstra, and Thibault, 2003) and theoretical values (Kouraet al., 2005). A phenomenological density dependence of theshell effects is introduced, assuming that these vanish at nsat.

TABLE III. (Continued)

ModelNuclear

interactionDegrees offreedom

Mmax,(M⊙)

R1.4M⊙ ,(km) Ξ

Publiclyavailable References

STOSQ209nπ TM1 n;p;α;ðA;ZÞ;π;q 1.85 13.6 0.21 No Nakazato, Sumiyoshi, andYamada (2008)

STOSQ162n TM1 n; p; α; ðA; ZÞ; q 1.54 No Nakazato, Sumiyoshi, andYamada (2013)

STOSQ184n TM1 n; p; α; ðA; ZÞ; q 1.36 � � �b No Nakazato, Sumiyoshi, andYamada (2013)

STOSQ209n TM1 n; p; α; ðA; ZÞ; q 1.81 14.4 0.20 No Nakazato, Sumiyoshi, andYamada (2008, 2013)

STOSQ139s TM1 n; p; α; ðA; ZÞ; q 2.08 12.6 0.26 Yes Sagert et al. (2012) andFischer, Klähn et al. (2014)

STOSQ145s TM1 n; p; α; ðA; ZÞ; q 2.01 13.0 0.25 Yes Sagert et al. (2012)STOSQ155s TM1 n; p; α; ðA; ZÞ; q 1.70 9.93 0.25 Yes Fischer et al. (2011)STOSQ162s TM1 n; p; α; ðA; ZÞ; q 1.57 8.94 0.26 Yes Sagert et al. (2009)STOSQ165s TM1 n; p; α; ðA; ZÞ; q 1.51 8.86 0.25 Yes Sagert et al. (2009)

aValues taken from Marek and Janka (2009).bMmax below 1.4M⊙.



For light nuclei, it incorporates the Pauli-blocking shifts ofTypel et al. (2010). Furthermore, light nuclei receive self-energy shifts originating from the mesonic mean fields. As anadditional phenomenological interaction, excluded-volumeeffects are applied for nucleons, light nuclei, and heavy nuclei.In addition to standard spherical nuclei, a bubble phase withlow-density holes inmatter of higher density is also considered.The FYSS EoS has been used to explore the effect of lightnuclei in CCSN simulations (Furusawa,Nagakura et al., 2013).

e. HS

The basic model of the HS EoS (Hempel and Schaffner-Bielich, 2010) (HS) belongs to the class of extended NSEmodels and describes matter as a “chemical”mixture of nucleiand unbound nucleons in NSE. Nuclei are treated as classicalMaxwell-Boltzmann particles, nucleons with RMF modelsemploying different parametrizations. Several thousandsof nuclei are considered, including light ones. Bindingenergies are taken either from experimental measurements(Audi, Wapstra, and Thibault, 2003) or from various theo-retical nuclear structure calculations (Möller et al., 1995;Lalazissis, Raman, and Ring, 1999; Geng, Toki, and Meng,2005). The latter are chosen such that they were calculated forthe same RMF parametrization as the one applied to nucleonsif available; otherwise, the data from Möller et al. (1995) areused. The following medium modifications are incorporatedfor nuclei: screening of the Coulomb energies by the sur-rounding gas of electrons in the Wigner-Seitz approximation,excited states in the form of an internal partition functionusing the level density of Fái and Randrup (1982), which isadapted from the NSE model of Ishizuka, Ohnishi, andSumiyoshi (2003), and excluded-volume effects. Note thatfurther explicit medium modifications of nuclei are notconsidered in HS. Since the description of heavy nuclei isbased on experimental nuclear masses, the HS EoS includesthe correct shell effects of nuclei in vacuum. On the otherhand, the use of nuclear mass tables limits the maximum massand charge numbers of nuclei (Buyukcizmeci et al., 2013).The first version Hempel and Schaffner-Bielich (2010) used

the RMF parametrization TMA (Toki et al., 1995). A fewaspects of the model have been changed in the later versions(Hempel et al., 2012), namely, a cutoff for the highestexcitation energy of nuclei is introduced, experimentalnucleon masses are used, and only nuclei left of the neutrondrip line are considered. At present, EoS tables are availablefor the following RMF parametrizations: TM1 (Sugahara andToki, 1994; Hempel et al., 2012), TMA (Toki et al., 1995;Hempel and Schaffner-Bielich, 2010; Hempel et al., 2012),FSUgold (Todd-Rutel and Piekarewicz, 2005; Hempel et al.,2012), NL3 (Lalazissis, König, and Ring, 1997; Fischer,Hempel et al., 2014), DD2 (Typel et al., 2010; Fischer,Hempel et al., 2014), and IUFSU (Fattoyev et al., 2010;Fischer, Hempel et al., 2014). We denote them by “HS(x),”where “x” indicates the nuclear interaction employed.

f. SFHo and SFHx

Two additional EoSs based on HS were published bySteiner, Hempel, and Fischer (2013), SFHo (“o” for optimal)and SFHx (“x” for extreme) with new RMF parametrizations

that were fitted to some NS radius determinations. These twoEoSs result in rather compact NSs and have moderately,respectively, very low values of the slope parameter of thesymmetry energy L (Steiner, Hempel, and Fischer, 2013;Fischer, Hempel et al., 2014; Hempel et al., 2015).

g. SHT(NL3), SHO(FSU), and SHO(FSU2.1)

The EoSs of Shen, Horowitz, and O’Connor (2011) andShen, Horowitz, and Teige (2011) are based on differentunderlying physical descriptions in different regimes ofdensity and temperature. Uniform nuclear matter at highdensities and temperatures is described by a RMF model.At intermediate densities, the same RMF model is used withincalculations of nonuniform matter, generating a representativeheavy nucleus and unbound nucleons, but no light nuclei. Atlow densities and temperatures, a special form of the VEoS isused which includes virial coefficients up to second orderamong nucleons and α particles. The VEoS is not usingFermi-Dirac statistics for nucleons, but only incorporatescorrections for it as part of the virial coefficients. Nuclei withmass numbers A ¼ 2 and 3 are not considered as explicitdegrees of freedom, but 8980 nuclei with mass number A ≥ 12are included. The contribution of heavy nuclei in NSE ismodeled as a noninteracting Maxwell-Boltzmann gas withoutconsidering excluded-volume effects. Coulomb screening isincluded for heavy nuclei, but not for α particles. Note thatα particles are present only in the virial part of the EoS, which iscompletely independent of the RMF interaction. The threedifferent prescriptions are merged to a single table by mini-mizing the free energy. In addition, a smoothing and inter-polation procedure is applied (Shen, Horowitz, and O’Connor,2011; Shen, Horowitz, and Teige, 2011).The EoSs of G. Shen et al. are available for two different

RMF interactions: NL3 (Shen, Horowitz, and Teige, 2011)[SHT(NL3)] and FSUgold (Shen, Horowitz, and O’Connor,2011). A density dependence of the scalar meson-nucleoncoupling was introduced below 5 × 10−3 fm−3 in case of theNL3 interaction in order to match the energy per nucleon of aunitary neutron gas (Shen, Horowitz, and Teige, 2010). Sincethe FSUgold parametrization leads to a maximum NS mass ofonly 1.7M⊙, an additional phenomenological pressure contri-butionwas introduced for densities above 0.2 fm−3, leading to asufficiently high maximum mass of 2.1M⊙. This EoS wascalled “FSU2.1” and we abbreviate it as SHO(FSU2.1), and theEoSwith the unmodified FSUgold parametrization SHO(FSU).

2. Including additional degrees of freedom

An EoS covering the whole thermodynamic parameterrange relevant for CCSNe and NS mergers should be ableto correctly describe cold β-equilibrated NSs. As discussed inSec. V.A.2, it might turn out that only nucleonic matter ispresent in cold NSs. However, this does not mean thatadditional degrees of freedom could not occur in stellarcore-collapse events and NS mergers, where matter is stronglyheated in addition to being compressed to densities abovenuclear matter saturation density. The temperatures anddensities reached can become so high that a traditionaldescription in terms of electrons, nuclei, and nucleons is nolonger adequate. Compared with the cold NS EoS,



temperature effects favor the appearance of additional par-ticles such as pions and hyperons and they become abundantin this regime. A transition to quark matter can also not beexcluded. In recent years, some models have been developedwhich extended existing purely nuclear models (as discussedin the previous section) by including pions, hyperons, orquarks.Let us start with the models including hyperons. Ishizuka

et al. (2008) and Oertel, Fantina, and Novak (2012) consid-ered the whole baryon octet. The former EoS is an extensionof the STOS EoS by Shen et al. (1998a) and the latter of theLS220 EoS (Lattimer and Swesty, 1991). Ishizuka et al.(2008) fixed the hyperonic interactions following a standardprocedure for RMF models. For the vector couplings, sym-metry constraints are imposed, assuming SUð6Þ flavor sym-metry following Schaffner and Mishustin (1996) in theisoscalar sector and isospin symmetry in the isovector one.The remaining scalar couplings of hyperons to nucleons areadjusted to reproduce standard values of the single-particlehyperonic potentials in symmetric nuclear matter at saturationdensity, extracted from hypernuclear data (see Sec. III.A.1).These single-particle potentials are given by (seeSec. III.B.2.a)

Uj ¼ Vj − Sj; ð38Þinvolving scalar and vector self-energies. Standard values forUΛ ¼ −30 MeV and UΞ ¼ −15 MeV are assumed, whereasthe situation for UΣ is ambiguous and Ishizuka et al. (2008)presented several versions of the EoS with UΣ ¼ −30, 0, 30,90 MeV (“STOSYxxx,” where “xxx” indicates the valueof the potential, and a prepended “A” if it is attractive).In the following discussion, we keep the version withUΣ ¼ þ30 MeV. Oertel, Fantina, and Novak (2012) addedhyperons by extending the model by Balberg and Gal (1997)to finite temperature. This model is a nonrelativistic potentialmodel similar to the one in Lattimer and Swesty (1991) for thenuclear part. The hyperonic couplings are readjusted to remaincompatible with the single-particle hyperonic potentials innuclear matter, but, at the same time, predict maximum NSmasses in approximate agreement with the measurements ofDemorest et al. (2010).Ishizuka et al. (2008) and Oertel, Fantina, and Novak

(2012) presented models, in which pions are also included.The former is denoted by “STOSYπxxx.” Pions are treated asan ideal Bose gas. Obviously, without interactions, π− willform a Bose condensate below some critical temperature,depending on the density, as discussed extensively (Migdalet al., 1990; Glendenning, 1997). It is now commonlyassumed that there is an s-wave πN repulsion, preventingpions from condensing. Ishizuka et al. (2008) showed thatindeed, adding an effective πN interaction, the domain intemperature and density where pions condense is stronglyreduced. The main effect of pions on the EoS occurs, however,at high temperature and the ideal gas should be a goodapproximation in this regime. A simplified version includingonly pions in the LS220 EoS is used by Peres, Oertel, andNovak (2013) (“LS220π”). Nakazato, Sumiyoshi, andYamada (2008) extended the STOS EoS in the sameway (“STOSπ”).

Subsequently different models including only Λ hyperonshave been developed. The first one is the work by Shen et al.(2011) (“STOSΛ”), which is very similar to the work byIshizuka et al. (2008), except that slightly different hyperoninteractions are employed. The motivation is that the Λrepresents, together with the Σ− hyperon, probably the mostimportant hyperonic degree of freedom in hot dense super-nova matter. Thus, including the Λ allows for discussinggeneral features of the effects coming from the hyperonicdegrees of freedom without the necessity of resolving thecomplicated particle composition in the presence of manydifferent hyperons. In addition, the ΛN and ΛΛ interactionsare the best constrained from experimental data. And, sinceless degrees of freedom are populated, the NS maximum massis less reduced from that of purely nucleonic EoSs. Anextended version of the LS220 EoS including only Λ hyperons(“LS220Λ”) has been discussed by Gulminelli et al. (2013)and Peres, Oertel, and Novak (2013). It has the feature that astrangeness driven first-order phase transition occurs at theonset of hyperons (Schaffner-Bielich and Gal, 2000;Schaffner-Bielich et al., 2002; Gulminelli, Raduta, andOertel, 2012; Oertel et al., 2016) (see Sec. III.D). Banik,Hempel, and Bandyopadhyay (2014) constructed two differ-ent extended versions of the HS statistical model includingΛ hyperons. The density-dependent RMF parametrizationDD2 is employed and the setup for the hyperonic couplingsis similar to the one used by Ishizuka et al. (2008), i.e.,assuming SUð6Þ symmetries for the vector couplings. Thevalue of the Λ single-particle potential in symmetric nuclearmatter determines the remaining scalar couplings. The basicmodel is denoted as BHBΛ and the version including addi-tional short-range repulsion in the ΛΛ channel by BHBΛϕ.The onset density for hyperons lies between 2 and 3 times

nsat at low temperatures. As expected, upon increasing thetemperature, the density domain where hyperons appear isenlarged, in particular, above 15–20 MeV; see Fig. 11. Notethat due to the presence of light nuclei in the BHBΛϕ EoS(Banik, Hempel, and Bandyopadhyay, 2014), hyperonsappear at a much higher temperature in a large density domainthan in the other models. Pions become more abundant at hightemperatures too. This can also be seen from Fig. 12. In thebottom panels, the different particle number fractions areshown as a function of temperature for nB ¼ 0.15 [Fig. 12(e)]and 0.3 fm−3 [Fig. 12(f)] and for Yq ¼ 0.1. The influence ofYq on the appearance of (neutral) Λ hyperons is less importantthan for charged particles. The more asymmetric the matter,the higher is the charge chemical potential, and the higher theabundance of charged particles. In neutron-rich matter thecharge chemical potential is negative, favoring negativelycharged particles. This is the reason why in NS matter, Σ− orΞ− can become enhanced with respect to Λ hyperons, even ifthey have a higher mass. Thermal effects alleviate theinfluence of the chemical potential (Ishizuka et al., 2008;Oertel, Fantina, and Novak, 2012). If the baryon numberdensity remains constant, the overall hyperon fractionsdecrease with increasing Yq (Prakash et al., 1997).Concerning the influence on thermodynamic properties,

pressure and energy density are shown as functions oftemperature for different models in the upper and middle



panels of Fig. 12. The STOS (Shen et al., 1998a), LS(Lattimer and Swesty, 1991), and HS(DD2) EoSs are com-pared with their corresponding versions including hyperons(Ishizuka et al., 2008; Shen et al., 2011; Gulminelli et al.,2013; Banik, Hempel, and Bandyopadhyay, 2014) and/orpions (Ishizuka et al., 2008; Nakazato, Sumiyoshi, andYamada, 2008; Peres, Oertel, and Novak, 2013). As seenfrom Fig. 12, the effect of the additional particles on thethermodynamic quantities is not negligible for high densityand temperature. The main effect is a reduction of pressuredue to the additional degrees of freedom.There exist EoSs in which the nuclear model of Shen et al.

(1998a) has been supplemented with a phase transition toquark matter at high density and temperature too. In Nakazato,Sumiyoshi, and Yamada (2008, 2013) and the work by Sagert

et al. (2009, 2012), Fischer et al. (2011), and Fischer, Klähnet al. (2014) the MIT bag model (Chodos et al., 1974; Farhiand Jaffe, 1984) is applied to the quark matter phase. Thetransition from hadronic matter to quark matter is described bya Gibbs construction in both cases (see Sec. III.D). In additionto the nuclear interaction, the parameters of the model are thebag constant B and the strange quark mass, and possibly thecoupling constant of strong interaction corrections. Theseparameters determine the densities for the onset of the quark-hadron phase coexistence region and the pure quark phase. Itcan be observed that the onset density for the mixed phase isnoticeably lowered with decreasing charge fraction. In par-ticular, this means that the critical density in asymmetricCCSN or NS matter can be lower than in symmetric matter.Furthermore, it was found that the critical density is

0

510

15

20

25

30

35

40

45

50T

(MeV

)

10-6 10-5 10-4 10-3 10-2 10-1 1

nB (fm-3)

Yq 0.1

LS220BHBSTOSSTOSY30

0

510

15

20

25

30

35

40

45

50

T(M

eV)

10-6 10-5 10-4 10-3 10-2 10-1 1

nB (fm-3)

Yq 0.3

LS220BHBSTOSSTOSY30

0

510

15

20

25

30

35

40

45

50

T(M

eV)

10-6 10-5 10-4 10-3 10-2 10-1 1

nB (fm-3)

Yq 0.5

LS220BHBSTOSSTOSY30

FIG. 11. The lines delimit regions in temperature and baryon number density where the number density fractions of Λ hyperons exceed10−4 for the different models. From left to right corresponds to a fixed hadronic charge fraction Yq ¼ 0.1, 0.3, and 0.5. Λ hyperonsappear at high densities and temperatures, i.e., in the upper right part of each panel.

0

5

10

15

20

p (M

eV/fm

3 )

(a)

HS(DD2)BHBΛ

BHBΛφ

20

30

40

50

60

70

(b)

STOSSTOSY30

STOSY30πSTOSΛ

0

0.1

0.2

0.3

0.4

ε/(m

p n B

) -

1

(c)

LS220LS220ΛLS220π

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0.1

0.2

(d)

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0.1

0.2

0.3

0 25 50 75 100

Ys,

Yπ

T (MeV)

(e) 0

0.1

0.2

0.3

0 25 50 75 100

T (MeV)

(f)

STOSY30π:YSSTOSY30π:Yπ

FIG. 12. Thermodynamic quantities as functions of temperature for nB ¼ 0.15 (left) and 0.3 fm−3 (right), corresponding roughly to 1and 2 times nuclear matter saturation density, and a charge fraction of Yq ¼ 0.1. The upper panels show the pressure, the middle ones thescaled internal energy per baryon with respect to the proton mass, and the lower ones the number fractions of additional particles. TheseareΛ hyperons for “STOSΛ,” “LS220Λ,” “BHBΛ,” and “BHBΛϕ”; all hyperons for “STOSY30” and “STOSY30π”; and the π− fractionfor “LS220π.” See the text for an explanation of the acronyms.



significantly reduced due to weak equilibrium with respect tostrangeness. A value close to nuclear saturation or even belowis thus not in contradiction with any terrestrial experimentfrom HICs. This fact is exploited in the model of Sagert et al.(2009), where the bag constant has been chosen such that thestrongly asymmetric matter in compact stars leads to almostpure quark stars with only a thin hadronic layer. These modelsare labeled as “STOSQxxxs,” where “xxx” indicates the valueof the bag constant B1=4 in MeV, and “s” stands for Sagertet al. The models of Nakazato, Sumiyoshi, and Yamada(2008) lie in a parameter range (B1=4 ¼ 209 MeV) where thecritical density is much higher, such that the resulting NSshave only a small quark core. Strong interaction correctionsare not used in this model. If a thermal pion gas is included inthe hadronic phase, the transition to quark matter occurs atconsiderably higher densities (Nakazato, Sumiyoshi, andYamada, 2008) due to the softening of the hadronic EoSby the pions. The latter models are labeled “STOSQxxxn” forthe one with quarks and “STOSQxxxnπ” for the one withquarks and pions. Nakazato, Sumiyoshi, and Yamada (2013)calculated additional hybrid EoS tables for B1=4 ¼ 162 and184 MeV. Because the corresponding maximum masses of1.54M⊙ and 1.36M⊙ are well below the observed pulsarmasses, in the following we consider only the table withB1=4 ¼ 209 MeV as a representative example of the hybridEoSs of Nakazato, Sumiyoshi, and Yamada (2008, 2013). Thethermodynamic properties are strongly influenced by thepossible existence of quark matter at high densities andtemperatures and, in particular, the phase transition can havean important effect on the dynamics of CCSNe (seeSec. VI.B.1).

3. Compatibility with experimental and observational constraints

In this section we compare the results of the general purposeEoSs with several constraints that have been introduced inSec. IV. As can be seen from Table IV, most of the employedinteractions give reasonable properties for compressibility,saturation density, and binding energy of symmetric matter,except that LS180 has a compressibility at the lower end of theallowed range and LS375 and TMA have values above theallowed range. However, some models give symmetry ener-gies and slopes far off the best present constraints (seeFig. 13), i.e., the dependence of the EoS on Yq is probablynot correctly described. DD2, SFHo, IUFSU, and FSUgold(2.1) are in the best agreement.NS masses (see Figs. 14, 15, and Table III) probably

represent the presently most reliable observational constrainton the compact star EoS. LS180 and the models based onFSUgold give too low masses and HS(IUFSU) is onlymarginally compatible. Note that the maximum mass dependsonly very little on the treatment of the inhomogeneous part ofthe EoS, such that all models with the same nuclear interactiongive essentially the same maximum mass. This is not the casefor the radii of intermediate-mass NSs, which are moresensitive to the treatment of the crust. Here slight differencescan be observed between HS(TM1), STOS, and FYSS, whichare all based on the TM1 interaction; see Fig. 14 and Table III.Since the H&W is not publicly available, in Fig. 14 the resultsof the model by Gulminelli and Raduta (2015) are plotted.

This model is based on the same Skyrme interaction SkA.Note that the radius of a 1.4M⊙ NS of Gulminelli and Raduta(2015) differs by about 1 km compared to the value of H&Was given in Table III. In Sec. IV we outlined the difficulties indetermining reliable NS radii and that presently no consensuson the allowed range of values can be obtained. Let usmention, however, that if small radii for 1.4M⊙ NSs of theorder 10–12 km were confirmed [as reported, e.g., by Özelet al. (2016)], then some of the hadronic models shown here

20

40

60

80

100

120

26 28 30 32 34 36 38 40 42

L (M

eV)

J (MeV)

LSDD2

FSUgoldSka

IUFSUNL3

SFHoSFHxTM1TMA

FIG. 13. The slope parameter of the symmetry energy L vs thevalue of the symmetry energy J at normal nuclear matter density.The light gray region is the constraint of Lattimer and Lim(2013), and the dark gray region is taken from Fig. 8. Thedifferent symbols show the values of the nucleon interactions ofTable IV that are applied in the general purpose EoSs of Table III.FSU2.1 gives the same value as FSUgold.

TABLE IV. Nuclear matter properties of the parametrizations forthe nuclear interaction used in the general purpose EoS of Table III.Listed are the saturation density nsat, binding energy Bsat, incom-pressibility K, skewness coefficient Q ¼ −K0, symmetry energy J,and symmetry energy slope coefficient L at saturation density at zerotemperature.

Nuclearinteraction

nsat(fm−3)

Bsat(MeV)

K(MeV)

Q(MeV)

J(MeV)

L(MeV)

SKa 0.155 16.0 263 −300 32.9 74.6LS180 0.155 16.0 180 −451 28.6a 73.8LS220 0.155 16.0 220 −411 28.6a 73.8LS375 0.155 16.0 375 176 28.6a 73.8TM1 0.145 16.3 281 −285 36.9 110.8TMA 0.147 16.0 318 −572 30.7 90.1NL3 0.148 16.2 272 203 37.3 118.2FSUgold 0.148 16.3 230 −524 32.6 60.5FSUgold2.1 0.148 16.3 230 −524 32.6 60.5IUFSU 0.155 16.4 231 −290 31.3 47.2DD2 0.149 16.0 243 169 31.7 55.0SFHo 0.158 16.2 245 −468 31.6 47.1SFHx 0.160 16.2 239 −457 28.7 23.2

aThe value for the symmetry energy J is different from thevalue of 29.3 MeV in Lattimer and Swesty (1991). Theycomputed J as the energy difference between neutron andnuclear matter, whereas we are calculating J as the secondderivative with respect to Yq for symmetric matter at nsat; see alsoSteiner, Hempel, and Fischer (2013).



could be excluded, in particular, those based on TM1, NL3,and LS375.As can be seen from Fig. 15 and the data given in Table III,

the NS maximum masses of most of the extended models withadditional non-nucleonic degrees of freedom are not compat-ible with a 2M⊙ star. BHBΛϕ, STOSπ, STOSQ139s, andSTOSQ145s are the only ones with NS maximum massesabove 1.97M⊙. As discussed, for hyperonic EoSs this isrelated to the hyperonic interactions used. Recent studies forcold NS EoSs overcome the maximum mass problem butthese models for the interaction have not yet been applied tocompute a complete EoS covering the whole range of temper-ature, hadronic charge fraction, and baryon density. Anyway,with increasing temperature the effect of the interactions

becomes less important and most models agree qualitativelyfor the particle composition (cf. Fig. 12), whether or notcompatible with a 2M⊙ NS.The situation becomes even more severe if additional

constraints are included in the benchmarking. In the rightpanel of Fig. 16 the experimental flow constraint ofDanielewicz, Lacey, and Lynch (2002) for the pressure as afunction of baryon number density in symmetric nuclearmatter is depicted. For neutron matter (left panel), theconstraint from the χEFT calculation of Hebeler et al.(2013) is shown (cf. Sec. IV.B). These constraints arecompared with the EoS of symmetric and neutron matterobtained at T ¼ 0 from the different models. It is obvious thatnone of the present models is perfectly compatible with theneutron matter results from Hebeler et al. (2013) belowsaturation density. However, the error band shown in the leftpanel of Fig. 16 is perhaps too small (see Fig. 6), renderingsome of the models at least marginally compatible, such asDD2, SFHo, or FSUgold. The LS180 and LS220 models arein reasonable agreement with the constraint for nB ≳ 0.1 fm−3

but give too low pressures at lower densities. The modelsSFHo and SFHx have been fitted to some NS radiusdeterminations giving radii around 12 km for 1.4M⊙(Steiner, Lattimer, and Brown, 2010; Steiner, Hempel, andFischer, 2013). In the extreme model SFHx, it was tried tomake these as small as possible, within the employed class ofRMF interactions. This is the reason why they have very low(SFHx) or moderate pressure (SFHo) for neutron matteraround nsat and correspondingly low and moderate valuesof L. Similar observations as for neutron matter can be madefrom the comparison of the flow constraint with the EoS ofsymmetric nuclear matter: Many present models for thegeneral purpose EoS seem to give a too large pressure atsuprasaturation densities.The neutron matter EoS is a crucial anchor point for the NS

EoS and thus also of great significance for NS mergers.However, for CCSNe, matter is generally more symmetric andnuclear clusters are an important contribution to the subsat-uration EoS. In Fig. 5 , several of the general purpose EoSsare included in a comparison with experimental data forcluster formation (see Sec. IV.A.4). The LS220 EoS shows anotable underproduction of α particles, and SHT(NL3)and SHO(FSU2.1) an overproduction at high temperatures.The other general purpose EoSs FYSS, SFHo, STOS, andHS(DD2), are more or less in reasonable agreement with theconstraint. The current experimental data do not allow one tomake further judgements about details of the medium mod-ifications of nuclear clusters, e.g., to distinguish classicalexcluded-volume effects from quantum statistical Pauli block-ing. For further discussion, see Hempel et al. (2015).To conclude, there is not a single general purpose EoS that is

compatible with all constraints, even though we consideredonly a few of them.However, from the purely nucleonicmodelsSFHo, HS(DD2), and SHO(FSU2.1) are at least approximatelyconsistent. From the EoS with additional degrees of freedom,only BHBΛϕ, which is also based on DD2, would beacceptable. Nevertheless these EoSs have further drawbacksand weaknesses: in the models based on HS [SFHo, HS(DD2),BHBΛϕ] the treatment of light nuclei is not as advanced as in

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3

10 12 14 16 18

M (

M)

R (km)

LS180

LS220

LS375

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HS(TMA)

SFHo

SFHx

HS(FSU)

HS(IUFSU)

HS(NL3)

SHT(NL3)

SHO(FSU)

SHO(FSU2.1)

SKa

FIG. 14. Mass-radius relations of spherically symmetric NSs forthe different EoSs that cover the full thermodynamic parameterrange and include only nucleonic degrees of freedom; cf. Table III.The two horizontal bars indicate the two recent precise NS massdeterminations, PSRJ1614-2230 (Demorest et al., 2010) (hatchedblue) and PSR J0348þ 0432 (Antoniadis et al., 2013) (yellow).The curve labeled “SkA,” although based on the same nuclearinteraction, does not represent the H&W EoS, but the model byGulminelli and Raduta (2015).

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8 10 12 14 16 18 20

M (

M)

R (km)

LS220ΛLS220π

BHBΛBHBΛφSTOSΛ

STOSY30

STOSY30πSTOSπ

STOSQ139s

STOSQ145s

STOSQ155s

STOSQ162s

STOSQ165s

STOSQ209n

STOSQ209nπ

FIG. 15. Same as Fig. 14 for EoS models including additionaldegrees of freedom. The onset of additional degrees of freedom isvisible as a change in the slope.



the quantum statistical model (see Sec. III.C.4) or generalizedrelativistic density functional (see Sec. III.C.5), and it does notinclude an explicit medium dependence of the nuclear bindingenergies of heavy nuclei, which one could extract fromnucleons-in-cell-calculations (see Sec. III.C.6). Other EoSs(e.g., FYSS or SHO and SHT), which are more advanced insome of these aspects, do not fulfill constraints for themaximum mass or L. The SHO(FSU2.1) is compatible withthe maximum mass constraint only because of an ad hocmodification of the pressure at high densities. Furthermore, ofall possible light nuclei only the α particle is included in thismodel. Detailed nucleons-in-cell calculations are used forheavy nuclei, but only at intermediate to high densities, andlight nuclei are not taken into account in this regime at all. Theusage of different prescriptions in different regimes of T and nBcan also lead to problems in the thermodynamically consistentconstruction of transitions. The state of the art in modeling thegeneral purpose EoS is thus not really satisfactory. There is stillneed for new general purpose EoSs that employ modern EDFs(or even beyond) with good nuclear matter properties, thattackle the problem of additional degrees of freedom at highdensities and temperatures, and that give a detailed descriptionof clustering at subsaturation densities.

VI. APPLICATIONS IN ASTROPHYSICS

A. Binaries and binary mergers

Coalescing relativistic binary systems containing compactobjects, either NSs or BHs, are interesting in the context of theEoS of dense and hot matter. They are likely to be importantsources of detectable gravitational waves (GW) by advancedLaser Interferometer Gravitational-WaveObservatory (LIGO),Virgo, and Kamioka Gravitational Wave Detector (KAGRA),possibly before 2020. NS-NS and NS-BHmergers are believedto produce short gamma-ray bursts (sGRB). In addition, theymay represent the major source for the main component ofheavy r-process elements in the Universe; see, e.g., the recentreviews by Shibata and Taniguchi (2011), Faber and Rasio(2012), and Rosswog (2015) and references therein. All three,the GW signal, the sGRBs, and the r-process abundances,contain information on the EoS.

During the late inspiral phase of both NS-NS and NS-BHsystems, NSs become tidally deformed to an extent thatdepends on the underlying EoS. Numerical models suggestthat the GW frequency is very sensitive to the tidal deformationand thus to the underlying EoS (Shibata and Taniguchi, 2011;Faber and Rasio, 2012). However, the rate of NS-BH inspiralsis uncertain (to date no such system has been observed), and thetidal effects from these systems are probably not visible fornext-generation detectors since they occur at too high frequen-cies outside the range where the detectors are most sensitive(Pannarale et al., 2011). On the contrary, after the first detectionof GW emission from a BH-BH merger (Abbott et al., 2016),there appears to be a good chance for binary NS mergers to bedetected in the near future, and the tidal deformability hasprobably a strong enough impact on theGWsignal (Read et al.,2013). Additional information on the EoS can be obtained fromthe postmerger phase, in cases where the EoS supports theformation of a hypermassiveNS. The frequencies ofNS normalmodes after the merger are sensitive to the EoS and visible inthe GW signal; see, e.g., Sekiguchi et al. (2011), Bausweinet al. (2012), and Takami, Rezzolla, and Baiotti (2014).Measurements of their frequencies could tightly constrainNS masses and radii since they are strongly correlated.Figure 17 illustrates the correlation between the dominantGW frequency in the postmerger phase, normalized to the totalmass of the binary system, and the radius of a cold nonrotatingNS with M ¼ 1.6M⊙. It could even be possible to give anestimate for the NS maximum mass (Bauswein, Baumgarte,and Janka, 2013; Bauswein, Stergioulas, and Janka, 2014;Bauswein and Stergioulas, 2015).Fryer et al. (2015) and Lawrence et al. (2015) proposed

another test of the EoS in binary NS mergers. Numericalmodels suggest that a sGRB is produced in such a merger onlyif a BH forms sufficiently fast. The fate of the core of theremnant, and thus the BH formation time, depends on themaximum mass supported, and thus on the EoS. Althoughother factors influence the maximum mass, for instance, thespin rate or the angular momentum distribution inside the core(Kastaun and Galeazzi, 2015), they suggested that it ispossible to relate the existence of a sGRB to the maximummass of a cold nonrotating NS. Assuming NS binary mergersto be the dominant source of a sGRB, a combined analysis of

0.001

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p (M

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neutron matter

LS180LS220LS375

TM1DD2

FSUgold2.1Hebeler et al. (2010)

LS180LS220LS375

TM1DD2

FSUgold2.1Hebeler et al. (2010)

1

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0.2 0.3 0.4 0.5 0.6 0.7 0.8

nB (fm-3)

symmetric nuclear matter

(b)

DanielewiczFSUgold

IUFSUNL3

SFHoSFHxTMASkA

FIG. 16. Pressure as a function of baryon density within the EoS models listed in Table III for symmetric nuclear matter (right) and pureneutron matter (left). The results are compared with the theoretical calculation for neutron matter from Hebeler et al. (2013) and theexperimental flow constraint from Danielewicz, Lacey, and Lynch (2002) for symmetric matter.



the observed burst rate and the total merger rate with GWdetectors would then allow for constraining the EoS. Adifferent model for a sGRB invokes a supramassive magnetaras the central engine instead of a BH. Fan, Wu, and Wei(2013) extracted the NS maximum mass within this scenariousing GRB afterglow observations, where the abrupt declineof the x-ray plateau is interpreted as the collapse to a BH.Compact binary mergers eject initially extremely neutron-

rich matter, and have therefore already very early beenidentified as possible sources of r-process elements in theUniverse (Lattimer and Schramm, 1974). Recent calculationsshow that the conditions are favorable for producing a robustr-process abundance pattern of heavy nuclei that is close to thesolar (Rosswog, 2015). The r-process production rates, thefinal abundances, and the amount of ejected material dependon the chemical composition and the thermodynamic con-ditions in the ejecta, and thus on the EoS; see, e.g., Bauswein,Goriely, and Janka (2013), Wanajo et al. (2014), andSekiguchi et al. (2015). The radioactive decay of the freshlyproduced r-process elements should produce an electromag-netic transient, called a kilonova or macronova. Recently, afirst candidate event has been reported, associated with thesGRB 130603B (Tanvir et al., 2013). Macronova signatureshave also been found for sGRB 060614 (Yang et al., 2015)and sGRB 050709 (Jin et al., 2016). Assuming an almostequal-mass NS-NS merger as a source of sGRB 130603B,Hotokezaka, Kyutoku et al. (2013) showed that an EoS givingR1.35 ≲ 13.5 km is preferred in order to match the inferredrelatively large values of ejecta masses and velocities. Thisresult, however, depends strongly on the initial mass ratio andthe type of merger. Although expected to occur much lessfrequently, a compact binary merger, NS-NS or NS-BH, withan initial mass ratio substantially different from unity naturallyproduces high ejecta masses and velocities (Oechslin, Janka,and Marek, 2007). Kawaguchi et al. (2016) analyzed sGRB130603B within this scenario and found that a NS radiusabove 11 km is favored. It might be possible to obtain reliableinformation about the underlying EoS from the final

nucleosynthesis outcome with more observations and moredetailed simulations.

B. Core-collapse supernovæ

1. Dynamics

The dynamics of CCSNe results from a complex interplaybetween hydrodynamics, neutrino transport, weak inter-actions, and the EoS. The general expectation, calledMazurek’s law, is that due to the strong feedback, a smallmodification of one of the ingredients does not qualitativelychange the dynamics (Lattimer and Prakash, 2000). However,the quantitative differences induced by different EoS can belarge enough to govern the presence or absence of anexplosion (Janka, 2012a; Suwa et al., 2013).Since in the early phase electron pressure dominates and

later on the collapse proceeds homologously, the dynamics ofthe infall epoch has only a mild direct dependence on thebaryonic part of the EoS. It is sensitive to the electron fractionYe and the entropy. Changes of Ye result from electroncaptures (EC) on nuclei and free protons and therefore dependon the composition, in particular, the abundances and the massand charge of the appearing nuclei. The mass of the inner coreat bounce Mic is roughly proportional to hY2

Lei, the mean

fraction of trapped leptons squared (Lattimer, Burrows, andYahil, 1985), which is fixed and given by Ye at the momentwhere neutrino trapping sets in.Many studies, including those employing a statistical model

for the EoS, use the single nucleus approximation within thesimplified EC rates from Bruenn (1985) in order to determinethe evolution of Ye. In this simple model, the reactionQ value,determining the phase space available for the capture reaction,is approximated by the difference in proton and neutronchemical potentials,

μ≡ −μQ ¼ μn − μp; ð39Þwhich strongly depends on the EoS. Roughly, the larger μQthe larger the electron-capture rate. This quantity is illustratedfor different EoSs in Fig. 18. An entropy per baryon of s ¼ 1

12 13 14 150.7

0.8

0.9

1

1.1

1.2

R1.6

[km]

f peak

/Mto

t [kH

z/M

sun]

2.4 Msun

2.7 Msun

3.0 Msun

FIG. 17. Relation between the dominant GW frequency in thepostmerger phase of a NS-NS binary merger event and the radiusof a nonrotating NS with a gravitational mass of M ¼ 1.6M⊙.The frequency has been normalized with respect to the total massof the system. Different symbols refer to different total masses,where equal-mass binaries are assumed. From Bauswein,Stergioulas, and Janka, 2016 with kind permission of TheEuropean Physical Journal (EPJ).

-25

-20

-15

-10

-5

0

5

10

0.25 0.3 0.35 0.4 0.45 0.5

μ Q [M

eV]

Ye

HS(DD2)STOSLS220

HS(TM1)

FIG. 18. A comparison of the values of charge chemicalpotential μQ for different EoSs and typical thermodynamicconditions in the infall epoch. The proton fraction has beenvaried at constant entropy per baryon s ¼ 1 and constant baryonnumber density nB ¼ 10−3 fm−3.



has been chosen, corresponding to a typical value before shockformation and a baryon number density of nB ¼ 10−3 fm−3. Itis evident that not the saturation properties of coldmatterwithinthe EoS are relevant to determine μQ, but the treatment ofinhomogeneous matter, i.e., nuclei. For instance, the differencebetween the model HS(TM1) and STOS, using the sameinteraction and having the same saturation properties, is muchlarger than between HS(DD2) and HS(TM1) which have verydifferent nuclear properties but share the same treatment ofnuclei. This was already pointed out by Lattimer and Prakash(2000), where it was shown within a simple analytical modelthat μQ was much more sensitive to surface energies than to thebulk symmetry energy.The EC rates also depend on entropy. Already at the stage

of the progenitor small differences in entropy arise betweenthe EoS employed for core collapse and the progenitormodel, in general based on a nuclear reaction network.In addition, in the STOS EoS, the entropy contribution ofthe thermal motion of heavy nuclei is missing (seeSec. V.D.1.c), reducing the entropy and thus underestimat-ing the deleptonization.Another effect is that within the description of Bruenn

(1985), the EC rate strongly decreases with the neutron numberN. Since in general the distribution of nuclear abundances islarge in statistical models, the average heavy nucleus can bevery different from the single heavy nucleus in the SNA (seeSec. III.C.2 and Fig. 10). In HS(TM1), for instance, N is ingeneral smaller than in STOS,where the averagemass of heavynuclei is overestimated (related to the Thomas-Fermi approxi-mation and the used value for the gradient energy coefficient,see Sec. V.D.1.c), leading to a higher EC rate in the domainwhere EC on nuclei is dominant (Hempel et al., 2012).A remark of caution is in order here. It was clearly

demonstrated that it is not sufficient to use one average heavynucleus in order to determine the global EC rate on nuclei,because of shell effects on individual rates which are notcorrectly reproduced within this approximation; see, e.g.,Langanke et al. (2003) and Juodagalvis et al. (2010). Bettercalculations for individual rates, attenuating, in particular, thestrong suppression of EC on neutron-rich nuclei present in the(Bruenn, 1985) prescription, change the infall evolution andthe bounce properties too. The individual EC rates can changeby an order of magnitude from one nucleus to another whichhas an important effect on the final YLe

; see, e.g., Hix et al.(2003), Furusawa, Nagakura et al. (2013), Raduta, Gulminelli,and Oertel (2016), and Sullivan et al. (2016). The impact on thedynamics of a full CCSN simulation is still to be seen.Concerning the direct impact of the EoS and the nuclear

composition, Hempel et al. (2012) performed a detailedanalysis of the different stages during the infall epochcomparing the LS180, STOS, and HS(TM1) EoSs confirmingwithin a simulation some of the previously discussed effects.At the time of bounce small differences can be observedbetween the different EoSs. An important effect is that a lowcentral Ye is correlated with a low core mass; see, e.g.,Sumiyoshi et al. (2005), Janka (2012a), Hempel et al. (2012),Steiner, Hempel, and Fischer (2013), and Suwa et al. (2013).A smaller core mass at bounce leads to a weaker shockforming closer to the center. Naively, this would lead to a

situation less favorable for an explosion. In addition, moremass overlays the core in this case rendering an explosion stillmore difficult. However, different effects compete. For in-stance, a stronger deleptonization leads to a higher neutrinoluminosity which in turn heats the shock more strongly.Therefore no clear statement is possible and most studiesin spherical symmetry conclude on a minor effect of the EoSon the overall dynamics (Sumiyoshi et al., 2005; Hempelet al., 2012; Janka, 2012a; Steiner, Hempel, and Fischer,2013; Suwa et al., 2013; Fischer, Hempel et al., 2014; Togashiet al., 2014). In particular, it is difficult to relate single nuclearmatter parameters of the EoS to CCSN dynamics, unless onecompares EoSs that show very pronounced differences. Thefact that the existing general purpose EoSs often differ inseveral properties, also because of correlations among differ-ent parameters (see Sec. IV), makes systematic investigationsdifficult. In addition, as illustrated, the treatment of inhomo-geneous matter, properties of nuclei, and thermal effects arefound to be equally important as the very neutron-rich anddense part of the EoS (Fischer, Hempel et al., 2014). Thismight be different for BH formation, where the PNS maxi-mum mass is decisive (cf. Sec. VI.B.3).A more compact and more rapidly contracting PNS

resulting from a “softer” EoS6 generally seems to befavorable for explosions in multidimensional simulations.In particular, neutrinos are emitted with higher fluxes andhigher energies (Marek, Janka, and Müller, 2009). This notonly enhances neutrino cooling in 2D, but favors theformation of more violent hydrodynamical instabilitiesand stronger convection (Janka, 2012a; Suwa et al.,2013). This is illustrated in Fig. 19, where the evolutionof the PNS radius (upper panel) and the shock radius(lower panel) is shown for three different EoSs within a 2Dsimulation by the Garching group. It is evident that in thiscase the EoS decides upon the explosion.There might be an imprint of the EoS on the neutrino and

GW signal. For instance, the faster deleptonization during theinfall epoch leads to an enhancement in the neutrino lumi-nosity at early times. The higher neutrino fluxes and higherenergies of the emitted neutrinos from a more compact PNSshould lead to differences in the neutrino spectra too. From agalactic supernova, these differences should indeed be observ-able with present detectors; see Sumiyoshi et al. (2005),Marek, Janka, and Müller (2009), and Suwa et al. (2013). Amore compact PNS leads to higher frequencies and largeramplitudes for the emitted GW too (Marek, Janka, and Müller,2009; Scheidegger et al., 2010). However, the EoS is not theonly varying parameter, for example, the unknown progenitorstructure or the treatment of neutrino transport can inducemodifications in the evolution of the CCSN, such that it seemsvery difficult to unambiguously identify one particular effect.Recent 1D studies of CCSNe with non-nucleonic degrees of

freedom focus on BH formation (see Sec. VI.B.3), where

6Please note that the terms “soft” and “stiff” for the EoS are notnecessarily related to the incompressibility of cold symmetric nuclearmatter or any other nuclear matter parameter due to thermal effects. Itmeans here simply that the overall PNS is more compressible for asoft EoS.



sufficiently high temperatures and densities are reached so thatthese degrees of freedom are expected to have a notable effect onthe dynamics. An exceptional case for regular supernovaexplosions, i.e., without BH formation, could be the onset ofnon-nucleonic degrees of freedom via a strong first-order phasetransition occurring close to saturationdensity.Aspointed out byGentile et al. (1993) and confirmed by Sagert et al. (2009) withdetailed Boltzmann neutrino transport, in spherically symmetricsimulations a second shock can be formed as a direct conse-quence of a phase transition to quark matter. This second shockwas found to be strong enough to unbind the outer layers once itmergeswith the standing accretion shock (Sagert et al., 2009). Inthis way, a CCSN explosion is triggered due to the phasetransition to quark matter. Furthermore, the passage of thesecond shock leads to a second neutrino burst that is dominatedby electron antineutrinos,measurablewith present-day detectors(Dasgupta et al., 2010).However, the NS maximum masses of the EoSs applied by

Sagert et al. (2009) (STOSQ162s, STOSQ165s) are wellbelow 2M⊙, and thus ruled out by NS observations. In thesubsequent works exploring this scenario [see, e.g., Sagertet al. (2010), Fischer et al. (2011, 2012), Nakazato,Sumiyoshi, and Yamada (2013), and Fischer, Hempel et al.(2014)], explosions could not be obtained if the maximummass of the employed EoS is sufficiently high. It is clear thatthe required stiffening in the quark phase to reach 2M⊙typically does not allow for a strong first-order phasetransition that seems to be necessary in this scenario to trigger

explosions. This can be related to the so-called masqueradeproblem (Alford et al., 2005), known for the mass-radiusrelation of NSs where quark matter could behave very similarto hadronic matter. On the other hand, it has not been shownyet that a SN explosion induced by a phase transition is ruledout by the latest pulsar mass measurements.Overall, there are still many uncertainties and open ques-

tions about the CCSN explosion mechanism, such as thedependency on the progenitor, effects of magnetic fields androtation, numerical convergence, the strength and scale ofintrinsic multidimensional hydrodynamic effects (turbulence,convection, SASI, etc.), or an accurate treatment of neutrinointeractions and their transport (Mezzacappa, 2005; Kotake,Sato, and Takahashi, 2006; Janka et al., 2007; Ott, 2009;Janka, 2012a; Burrows, 2013). The EoS is one of them. Itsrole is not yet fully understood, partly due to the highcomplexity of the system and the interplay of all the afore-mentioned aspects.

2. PNSs, neutrino-driven winds, and nucleosynthesis

A neutrino-driven wind (NDW) is the emission of a low-density, high entropy baryonic gas from the surface of a newlyborn PNS in a CCSN. It is driven by energy deposition ofneutrinos emitted from deeper layers and sets in after thelaunch of the SN explosion. It remains active in the firstseconds up to minutes. The NDW is of great importance forthe nucleosynthesis of heavy elements, as it has beenconsidered as one of the most promising sites for the r process;see, e.g., the review by Arcones and Thielemann (2013).However, previous sophisticated long-term simulations ofCCSNe (Fischer et al., 2010; Hüdepohl et al., 2010) haveshown that the matter emitted in the NDW is generally protonrich, allowing only for the so-called νp process (Fröhlichet al., 2006; Fröhlich, Martínez-Pinedo et al., 2006; Arcones,Janka, and Scheck, 2007; Roberts, Woosley, and Hoffman,2010; Arcones and Thielemann, 2013), which is not able toproduce the most heavy nuclei.Martínez et al. (2012), Roberts, Reddy, and Shen (2012),

and Roberts (2012) realized that these long-term simulationsof the PNS deleptonization phase and the NDW neglected theeffect of nuclear interactions in the charged-current (CC)interaction rates of neutrinos with unbound nucleons. Asoutlined in Sec. III.B.2, the nucleon single-particle energieswithin mean-field models can be written as the sum of akinetic part which has the form of a free gas, depending on aneffective mass, and an interaction potential Vi,

Ei ¼ Ekinðm�i Þ þ Vi: ð40Þ

The difference of neutron and proton energies, which deter-mines the energy available for (anti)neutrinos from CCreactions, depends on ΔV ¼ Vp − Vn. In asymmetric matterwithin the hot PNS, ΔV can be as large as 50 MeV. It does notaffect neutral current reactions, unless ΔV carries an addi-tional energy dependence and one has an inelastic reaction.The simulations of Martínez-Pinedo et al. (2012), Roberts,

Reddy, and Shen (2012), and Roberts (2012) found thatincluding Vi correctly modifies the evolution of neutrinospectra and the deleptonization of the PNS. ΔV induces an

0 100 200 300 400 5000

20

40

60

80

time [ms]

neut

ron

star

radi

us[k

m]

L&SH&WShen

0 100 200 300 400 5000

500

1000

1500

time [ms]

shoc

kra

dius

[km

]

L&SH&WShen

FIG. 19. The top panel shows the evolution of the PNS’s radiusfor a two-dimensional CCSN simulation employing the LS180,the H&W, and the STOS EoSs. The bottom panel shows theevolution of the shock radius. From Janka, 2012b.



increase of the antineutrino energies and a decrease of theneutrino energies. This leads to slightly neutron-rich con-ditions in the NDW. BecauseΔV is related to the potential partof the symmetry energy (Hempel et al., 2015), (anti)neutrinospectra and the conditions in the NDW are sensitive to theisospin dependence of the EoS.Although the correct treatment of mean-field effects in the

CC reactions favors less proton-rich conditions in the NDWthan previously determined, the values of 0.46 < Ye < 0.5obtained in the simulations7 of Roberts, Reddy, and Shen(2012) and Martínez-Pinedo, Fischer, and Hüther (2014) withdifferent EoSs are still not low enough for a robust r-processnucleosynthesis. Similar conclusions have been obtainedwithin the QCD phase transition scenario (see Sec. VI.B.1),where again a slightly proton-rich NDW is produced, leadingto a weak r process (Nishimura et al., 2012). However, thereare still many open questions, e.g., the employed approx-imations in the neutrino transport and reaction rates, a possibleprogenitor dependence, or the role of light nuclei in theenvelope of PNSs.For instance, Arcones et al. (2008) found that the envelope

composition is dominated by nucleons, deuterons, tritons,and α particles, in agreement with other works; see, e.g.,Sumiyoshi and Röpke (2008), Hempel et al. (2012), andFischer, Hempel et al. (2014). Arcones et al. (2008) showedthat light nuclei other than the α particle (mainly deuteronsand tritons) lead to a small reduction of the average energy ofthe emitted electron neutrinos.The long-term deleptonization and cooling of the PNS

(Prakash et al., 1997; Pons, Steiner et al., 2001) containsfurther interesting aspects related to the EoS. Within thedelayed BH formation scenario, the loss of thermal energyand, in particular, the deleptonization destabilizes the PNS(cf. Sec. VI.B.3). For an exploding CCSN, convection in thesubsequently contracting and cooling PNS is very sensitive tothe EoS, as in the early postbounce phase. Since convectiondepends strongly on the variation of pressure with leptonfraction YLe

at constant nB, the EoS dependence is mainlycharacterized by the symmetry energy (Roberts et al., 2012).They showed that the symmetry energy leaves an imprint inthe neutrino count rates of present-day neutrino detectors for agalactic CCSN.In the simulation of Suwa (2014) the long-term evolution of

the PNS was followed in a self-consistent manner startingfrom its birth in the CCSN up to ∼70 s. It is the first time thatsuch a simulation entered the regime of conditions where theformation of the crust is expected.

3. Black hole formation

In a stellar core-collapse event a NS is formed if theexploding star successfully unbinds the ejected material afterbounce. In a so-called failed CCSN, the outcome may equallybe a stellar mass BH if the expanding shock is not able tobreak through the infalling material and accretion pushes the

PNS over its mass limit on the time scale of seconds.Alternatively, there can be a delayed BH formation process,where either the cooling PNS becomes unstable or the fallback of ejecta causes the collapse to a BH in the minutesfollowing the bounce. Numerical studies of BH formation incore collapse have a long history; see, e.g., O’Connor and Ott(2011) and references therein.All scenarios have in common that the formation of an

apparent horizon is accompanied by a significant drop inneutrino luminosity since most of the neutrino emittingmaterial is swallowed up by the BH. The GW signal couldbe interesting too in this context, being sensitive to oscillationsin the hot PNS (Cerdá-Durán et al., 2013) and thus to theEoS. Pons, Steiner et al. (2001) and Nakazato et al. (2010)demonstrated that for a galactic event the time betweenbounce and BH formation tBH is possibly observable fromthe neutrino signal in the Super-Kamiokande detector.However, the evolution of the core collapse and tBH cruciallydepends also on the progenitor structure. O’Connor and Ott(2013) showed that it might be possible to get informationabout the compactness of the core of the progenitor star fromthe neutrino spectra and luminosities, which would allow oneto disentangle the effects of the progenitor and of the EoS tosome extent. In addition, rotation can strongly change not onlythe time until BH formation, but also the neutrino signal itself(Sekiguchi and Shibata, 2011). In particular, neutrino emis-sion continues on a reduced level well after BH formationfrom the newly formed accretion disk, rendering the inter-pretation of the neutrino signal less obvious.The EoS, as well, strongly influences the time until BH

formation, since it determines the maximum mass supportedby the hot PNS. There are two different physical mechanismsleading to the final gravitational instability: either the collapseis accretion induced or due to deleptonization and/or cooling.In the latter case, it is not necessarily the reduced thermal

pressure which destabilizes the cooling PNS, but deleptoni-zation. In the hot PNS, due to the presence of trappedneutrinos, matter is very lepton rich and Ye can be as highas 0.4 (Prakash et al., 1997; Pons et al., 1999). This leads to asuppression of additional degrees of freedom containingstrangeness (Prakash, Cooke, and Lattimer, 1995), such ashyperons, a kaon condensed phase, and/or a delayed phasetransition to quark matter (see also Sec. V.D). Consequently,metastable PNSs could exist, whose maximum mass is abovethat of cold, β-equilibrated NS (Prakash et al., 1997). Duringthe deleptonization of the hot PNS, the fraction of hyperons orquarks increases, eventually inducing a loss of stability and acollapse to a BH; see, e.g., Keil and Janka (1995), Baumgarteet al. (1996), and Pons, Steiner et al. (2001).For an accretion-induced collapse in a failed CCSN, tBH is

too short for considerable deleptonization. Here the PNScannot support the additionally accreted mass. The sensitivityof tBH to the EoS has been demonstrated in many studies; see,e.g., Sumiyoshi, Yamada, and Suzuki (2007), Fischer et al.(2009), O’Connor and Ott (2011), and Suwa et al. (2013).However, there is no straightforward relation between anyproperty of a given EoS, for instance, the maximum mass of acold β-equilibrated NS, and the PNS mass at the onset of BHcollapse. The reason is that BH formation is a dynamicalprocess, and the temperature, density, and Ye distribution in

7Note that the Ye values are different compared with the valuesreported by Roberts, Reddy, and Shen (2012), due to a previouscomputational error which was later corrected (Roberts, presentationat the MICRA workshop in Trento, 2013).



the hot PNS depends on many factors and is, in particular,very different from that of a cold and β-equilibrated NS.Hempel et al. (2012) and Steiner, Hempel, and Fischer (2013)proposed an interesting ansatz: employing an extensive set ofnuclear EoSs in simulations with a 40M⊙ solar metallicity(Woosley and Weaver, 1995) progenitor, it was shown that forthe given setup, tBH can be correlated with the maximum massof a β-equilibrated isentropic PNS at s ¼ 4. However, theevolution of a CCSN does not depend only on the EoS, but onmany other factors (as discussed previously) which makes itdifficult to unambiguously relate tBH to the EoS.Failed CCSNe have larger accretion rates than their

exploding counterparts, such that higher temperatures anddensities are reached within the PNS. As discussed inSec. V.D, this could lead to a sustained production of addi-tional degrees of freedom such as quarks or hyperons.Subsequently, the EoS is softened, supporting less massand reducing tBH compared with a purely nuclear EoS; see,e.g., Ishizuka et al. (2008), Sumiyoshi et al. (2009), Nakazato,Sumiyoshi, and Yamada (2010), Nakazato et al. (2012),Peres, Oertel, and Novak (2013), and Char, Banik, andBandyopadhyay (2015). Since these non-nucleonic degreesof freedom appear only deep inside the PNS, apart from tBH noconsiderable difference in the neutrino signal is to be expectedwith respect to a purely nuclear EoS, except if the appearanceof additional particles is accompanied by a phase transition(Nakazato, Sumiyoshi, and Yamada, 2010; Peres, Oertel, andNovak, 2013).Thus, although it is a promising field, there is still work

needed before we can conclude from the neutrino signal on theEoS. We emphasize again that such difficulties are rathertypical for observables of CCSNe (cf. Sec. VI.B.1).

VII. SUMMARY AND CONCLUSIONS

Describing properties of matter in compact stars, theirformation and merger processes is a very challenging task.The wide range of densities, temperatures, and chargefractions to be covered includes extreme values out of reachin terrestrial experiments. Therefore, one has to rely ontheoretical modeling. However, dense hadronic and quarkmatter is difficult to describe since the many-body problemwith strongly interacting particles has to be solved. In thisreview, we discussed theoretical and phenomenologicalapproaches to address these difficulties.In addition, we reviewed constraints on the EoS that have

been obtained from experiments, astrophysical observations,and ab initio calculations. Let us mention here some particu-larly important constraints. First, the recent observation of twoNSs with precisely and reliably determined masses of about2M⊙ has triggered intensive discussions on the compositionof matter in the central part of NSs. These results put strongconstraints on the high-density, low-temperature part of theEoS. Second, considerable progress has been made in recentyears concerning theoretical ab initio calculations of pureneutron matter up to roughly saturation density, thus con-straining the neutron-rich part of the EoS in this densityregime. Third, laboratory experiments are beginning to con-verge to a common prediction for the symmetry energy and itsslope around the saturation density.

There exist plenty of EoSs for cold NSs. To a lesser extentthis still holds for EoSs for homogeneous hot matter in PNSs.In this review, the emphasis has been put on EoSs that coverthe entire range of thermodynamic variables, which is relevantfor simulations of CCSNe and compact binary mergers. Theyare much more rare, although in recent years much effort hasbeen devoted to the development of new models, focusing ontwo aspects. First, the treatment of cluster formation andinhomogeneous matter at low densities and temperatures hasbeen considerably improved. It was realized that light nuclei,which were previously ignored, can be important. SubsequentCCSN simulations have shown that differences in the clusterdescription induce differences in the dynamical evolutionwhich are as important as those arising from different nuclearinteraction models. Second, improved interactions and addi-tional particles have been considered for the high density andtemperature part, such as hyperons, mesons, and quarks.These non-nucleonic degrees of freedom influence, in par-ticular, black hole formation, and NS-NS and NS-BHmergers.Despite all efforts, there is much room for improvement.

The cluster treatment is often based on a purely phenomeno-logical description with several approximations and simplifi-cations (see Sec. III.C). The interaction models employedcannot be considered definite. For instance, no presentlyexisting model is consistent with all available constraints.However, it is clear that some of the constraints have to beregarded with care. Not all of them have the same reliability asthe 2M⊙ NSmass measurements; see the discussion in Sec. IV.The quality of constraints is expected to improve in the

future. For example, the current efforts to determine NS radiiwith an unprecedented 5% precision by projects such asATHENAþ, NICER, LOFT, and others promise rich infor-mation regarding the inner NS structure. GW astronomy hasthe potential to give new and completely independent insightinto compact stars and their underlying EoS. New laboratoryexperiments and experimental facilities such as RIKEN,FRIB, FAIR, or NICA will provide new constraints for highdensity matter. We emphasize that all available generalpurpose EoSs are based on phenomenological approachesdue to the computational and conceptual complexity of moremicroscopic methods. In the future, the increase in computa-tional power is likely to allow the latter to provide EoSssuitable for astrophysical simulations too.To conclude this review, let us mention, without claiming to

be exhaustive, some important questions, which have to beaddressed to develop a more realistic EoS. Because of thelarge range of variables covered in simulations, very differentdomains are encountered.

• Can we obtain a reliable description of all basic baryonicfew-body interactions?

• How and under which conditions do non-nucleonicdegrees of freedom appear?

• When does nuclear matter deconfine?• Can we develop a QCD-based framework that covers therelevant range of variables?

• How do we better treat spatially inhomogeneous matterand cluster formation?

• How do we describe phase transitions consistently?



Answers to any of these problems will result in bettermodels for the astrophysical EoS and will help to understandvarious fundamental phenomena such as the composition anddynamics of NSs, the explosion mechanism of CCSNe, thethreshold to BH formation, the nature of gamma-ray bursts, orthe origin of heavy elements and the related galacticalchemical evolution. In turn, these astrophysical insights arepotentially relevant for the analysis of relativistic HICs andpossibly for the search of QCD phase transitions.

ACKNOWLEDGMENTS

We thank S. Furusawa, C. Ishizuka, K. Nakazato, and K.Sumiyoshi for providing the data of their EoS, T. Fischer forproviding Fig. 1, M. Fortin for providing Fig. 7, and A. Peregofor providing Fig. 2. We are grateful to A. Botvina, N.Buyukcizmeci, I. Mishustin, and A. Raduta for providing uswith the data for Fig. 10. We appreciate the thoughtful andencouraging comments of T. Fischer, T. Gaitanos, C.Miller, G.Röpke, J. Schaffner-Bielich, and H. Wolter to early versions ofthis review. This work was partially funded by NewCompStar,COSTAction MP1304. M. O. received financial support fromthe Agence Nationale de la Recherche Project SN2NSNo. ANR-10-BLAN-0503. M. H. is supported by the SwissNational Science Foundation. T. K. is grateful for support fromthe Polish National Science Centre (NCN) under GrantNo. UMO-2013/09/B/ST2/01560. S. T. is supported by theHelmholtz Association (HGF) through the NuclearAstrophysics Virtual Institute (No. VH-VI-417) and by theDFG through Grant No. SFB1245.

APPENDIX: RESOURCES

1. EoS databases

Here we present a list of publicly available EoS databases.Many authors provide their EoSs online, among them many ofthose presented in Sec. V.D.

• Lattimer and Swesty (1991)http://www.astro.sunysb.edu/dswesty/lseos.htmlThe original LS EoSs are available in the form of a

computer program. Several authors have generatedtabulated versions from it that can be found in the otherdatabases. The various tables can differ in the range ofthermodynamic variables covered and the details of theunderlying calculation.

http://www.astro.sunysb.edu/lattimer/EOS/main.htmlFour unpublished general purpose EoSs are tabulated

at this web site, where, according to their names, three ofthem are based on the Skyrme interactions SKI’, SKa,and SKM*, and the fourth has an incompressibility of370 MeV.

• Sumiyoshi (1998)http://user.numazu‑ct.ac.jp/~sumi/eos/index.html[Hosts data of Ishizuka et al. (2008) as well].

• Shen and Horowitz (2010)http://cecelia.physics.indiana.edu/gang_shen_eos/

• Hempel (2011)http://phys‑merger.physik.unibas.ch/~hempel/

eos.html

The resources listed provide data in the full space oftemperature, density, and asymmetry. A Web site that offersEoS tables for NS matter is as follows:

• Potekhin and Haensel (2013)http://www.ioffe.ru/astro/NSG/nseoslist.html

Different groups maintain online EoS collections withadditional features. Here we provide the links with a smallsynopsis as found at the corresponding Web sites:

• STELLARCOLLAPSE.ORGO’Connor and Ott (2008)http://www.stellarcollapse.org“..., a website aimed at providing resources supporting

research in stellar collapse, core-collapse supernovae,neutron stars, and gamma-ray bursts.”

STELLARCOLLAPSE.ORG provides not only tabulatedEoS data but hosts valuable resources, information,and freely available open source code for stellarcollapse and related phenomena. The available opensource codes are listed in Sec.2 of the Appendix.

• COMPOSE

Klähn, Oertel, and Typel (2013) and Typel, Oertel,and Klähn (2015)

http://compose.obspm.fr“The online service COMPOSE provides data tables for

different state-of-the-art equations of state (EoSs) readyfor further usage in astrophysical applications, nuclearphysics and beyond.”

COMPOSE has been developed by us with supportof the ESF-funded network CompStar, and the succes-sive COST Action MP1304, NewCompStar. The com-munity behind these research networks consists ofEoS developers and users. A driving idea behindCOMPOSE is not only to host a wide range of EoS databut to provide it in a flexible, multiple purpose, andreusable data format applicable for NS and CCSN EoSs.The LORENE library (Gourgoulhon et al., 2016) isCOMPOSE compatible.

• EOSDB

Ishizuka et al. (2015)http://aspht1.ph.noda.tus.ac.jp/eos/index.html“Our aim is to summarize and share the current

information on nuclear EoS which is available todayfrom theoretical/experimental/observational studies ofnuclei and dense matter.”

EOSDB offers the possibility to search, compare, andgraphically represent nuclear matter EoSs and relatedquantities online.

2. Open source simulation software

EoSs are a crucial input to many astrophysical simulations.We present a list of publicly available codes treating problemsrelated to this review.

• LORENE

Gourgoulhon et al. (2016)http://www.lorene.obspm.fr“... a set of C++ classes to solve various problems

arising in numerical relativity, and more generally incomputational astrophysics.”



http://dx.doi.org/http://www.astro.sunysb.edu/dswesty/lseos.html





http://dx.doi.org/http://www.astro.sunysb.edu/lattimer/EOS/main.html

http://dx.doi.org/http://user.numazu-ct.ac.jp/%7Esumi/eos/index.html





http://dx.doi.org/http://cecelia.physics.indiana.edu/gang_shen_eos/




http://dx.doi.org/http://phys-merger.physik.unibas.ch/%7Ehempel/eos.html






http://dx.doi.org/http://www.ioffe.ru/astro/NSG/nseoslist.html




http://dx.doi.org/http://www.stellarcollapse.org



http://dx.doi.org/http://compose.obspm.fr



http://dx.doi.org/http://aspht1.ph.noda.tus.ac.jp/eos/index.html







http://dx.doi.org/http://www.lorene.obspm.fr




• RNS

Stergioulas (1996)http://www.gravity.phys.uwm.edu/rns/“... constructs models of rapidly rotating, relativistic,

compact stars using tabulated equations of state whichare supplied by the user.”

• STELLARCOLLAPSE.ORGO’Connor and Ott (2008)http://www.stellarcollapse.orgoffers several codes:

(1) GR1D:spherically symmetric code for stellar collapse toneutron stars and black holes.

(2) GR1DV2:spherically symmetric neutrino radiationhydrodynamics.

(3) SNEC:the SuperNova explosion code

(4) CCSNMULTIVAR:multivariate regression analysis of gravitational wavesfrom rotating core collapse.

• AGILE-IDSALiebendörfer (2011)https://physik.unibas.ch/~liebend/download“... provides tools to run a rudimentary and approxi-

mate model of a core-collapse supernova with neutrinotransport in spherical symmetry through the phasesof stellar collapse, bounce, and early postbounceevolution.”

REFERENCES

Abbott, B. P., et al. (Virgo Collaboration and LIGO ScientificCollaboration), 2016, Phys. Rev. Lett. 116, 061102.

Abrahamyan, S., et al., 2012, Phys. Rev. Lett. 108, 112502.Ademard, G., et al. (INDRA Collaboration), 2014, Eur. Phys. J. A50, 33.

Agnello, M., et al. (FINUDA Collaboration), 2012, Phys. Rev. Lett.108, 042501.

Agrawal, B., S. Shlomo, and V. K. Au, 2005, Phys. Rev. C 72,014310.

Agrawal, B. K., J. N. De, and S. K. Samaddar, 2012, Phys. Rev. Lett.109, 262501.

Agrawal, B. K., J. N. De, S. K. Samaddar, M. Centelles, and X.Viñas, 2014, Eur. Phys. J. A 50, 19.

Ahn, J., et al., 2006, Phys. Lett. B 633, 214.Aichelin, J., 1991, Phys. Rep. 202, 233.Aichelin, J., and J. Schaffner-Bielich, 2009, in Relativistic Heavy-IonPhysics Vol. I/23 of Landolt-Börnstein, edited byR. Stock (Springer,Heidelberg).

Akmal, A., V. R. Pandharipande, and D. G. Ravenhall, 1998, Phys.Rev. C 58, 1804.

Alcain, P., P. A. Giménez Molinelli, J. Nichols, and C. Dorso, 2014,Phys. Rev. C 89, 055801.

Alcain, P. N., P. A. Giménez Molinelli, and C. O. Dorso, 2014, Phys.Rev. C 90, 065803.

Alcock, C., E. Farhi, and A. Olinto, 1986, Astrophys. J. 310, 261.Alexander, G., et al., 1968, Phys. Rev. 173, 1452.Alford, M., M. Braby, M. Paris, and S. Reddy, 2005, Astrophys. J.629, 969.

Alford, M., and S. Reddy, 2003, Phys. Rev. D 67, 074024.

Alford, M., et al., 2007, Nature (London) 445, E7.Alford, M. G., S. Han, and M. Prakash, 2013, Phys. Rev. D 88,083013.

Alford, M. G., K. Rajagopal, S. Reddy, and A.W. Steiner, 2006,Phys. Rev. D 73, 114016.

Alford, M. G., A. Schmitt, K. Rajagopal, and T. Schäfer, 2008, Rev.Mod. Phys. 80, 1455.

Alkofer, R., and L. von Smekal, 2001, Phys. Rep. 353, 281.Allton, C., et al., 2002, Phys. Rev. D 66, 074507.Alm, T., B. L. Friman, G. Röpke, and H. Schulz, 1993, Nucl. Phys. A551, 45.

Andronic, A., P. Braun-Munzinger, J. Stachel, and M. Winn, 2012,Phys. Lett. B 718, 80.

Angeli, I., and K. Marinova, 2013, At. Data Nucl. Data Tables99, 69.

Angeli, I., et al., 2009, J. Phys. G 36, 085102.Antić, S., and S. Typel, 2015, Nucl. Phys. A 938, 92.Antoniadis, J., et al., 2013, Science 340, 448.Aoki, S., et al., 1991, Prog. Theor. Phys. 85, 1287.Aoki, S., et al., 1995, Phys. Lett. B 355, 45.Aoki, S., et al. (HAL QCD Collaboration), 2012, Prog. Theor. Exp.Phys. 2012, 01A105.

Arcones, A., H.-T. Janka, and L. Scheck, 2007, Astron. Astrophys.467, 1227.

Arcones, A., and F.-K. Thielemann, 2013, J. Phys. G 40,013201.

Arcones, A., et al., 2008, Phys. Rev. C 78, 015806.Arndt, R. A., W. J. Briscoe, I. I. Strakovsky, and R. L. Workman,2007, Phys. Rev. C 76, 025209.

Arndt, R. A., I. I. Strakovsky, and R. L. Workman, 1994, Phys. Rev.C 50, 2731.

Audi, G., A. H. Wapstra, and C. Thibault, 2003, Nucl. Phys. A 729,337.

Avancini, S., C. Barros, D. Menezes, and C. Providência, 2010, Phys.Rev. C 82, 025808.

Avancini, S., J. Marinelli, D. Menezes, M. de Moraes, and C.Providência, 2007a, Phys. Rev. C 75, 055805.

Avancini, S., J. Marinelli, D. Menezes, M. de Moraes, and C.Providência, 2007b, Int. J. Mod. Phys. E 16, 2989.

Avancini, S., J. Marinelli, D. Menezes, M. Moraes, and A. Schneider,2007c, Phys. Rev. C 76, 064318.

Avancini, S., et al., 2008, Phys. Rev. C 78, 015802.Avancini, S., et al., 2009, Phys. Rev. C 79, 035804.Avancini, S., et al., 2012, Phys. Rev. C 85, 035806.Avancini, S. S., S. Chiacchiera, D. P. Menezes, and C. Providência,2010, Phys. Rev. C 82, 055807.

Aymard, F., F. Gulminelli, and J. Margueron, 2014, Phys. Rev. C 89,065807.

Baardsen, G., A. Ekström, G. Hagen, and M. Hjorth-Jensen, 2013,Phys. Rev. C 88, 054312.

Baiko, D., and P. Haensel, 1999, Acta Phys. Pol. B 30, 1097.Baiko, D., A. Potekhin, and D. Yakovlev, 2001, Phys. Rev. E 64,057402.

Balberg, S., and A. Gal, 1997, Nucl. Phys. A 625, 435.Baldo, M., I. Bombaci, and G. F. Burgio, 1997, Astron. Astrophys.328, 274.

Baldo, M., and F. Burgio, 2001, Lect. Notes Phys. 578, 1.Baldo, M., and G. Burgio, 2012, Rep. Prog. Phys. 75, 026301.Baldo, M., G. F. Burgio, M. Centelles, B. K. Sharma, and X. Viñas,2014, Phys. At. Nucl. 77, 1157.

Baldo, M., L. Ferreira, and O. Nicotra, 2004, Phys. Rev. C 69,034321.

Baldo, M., and L. S. Ferreira, 1999, Phys. Rev. C 59, 682.



http://dx.doi.org/http://www.gravity.phys.uwm.edu/rns/








http://dx.doi.org/https://physik.unibas.ch/%7Eliebend/download



http://dx.doi.org/10.1103/PhysRevLett.116.061102


http://dx.doi.org/10.1140/epja/i2014-14033-x




http://dx.doi.org/10.1103/PhysRevC.72.014310




http://dx.doi.org/10.1140/epja/i2014-14019-8

http://dx.doi.org/10.1016/j.physletb.2005.12.057

http://dx.doi.org/10.1016/0370-1573(91)90094-3






http://dx.doi.org/10.1086/164679

http://dx.doi.org/10.1103/PhysRev.173.1452

http://dx.doi.org/10.1086/430902

http://dx.doi.org/10.1086/430902

http://dx.doi.org/10.1103/PhysRevD.67.074024

http://dx.doi.org/10.1038/nature05582






http://dx.doi.org/10.1016/S0370-1573(01)00010-2


http://dx.doi.org/10.1016/0375-9474(93)90302-E

http://dx.doi.org/10.1016/0375-9474(93)90302-E


http://dx.doi.org/10.1016/j.adt.2011.12.006


http://dx.doi.org/10.1088/0954-3899/36/8/085102

http://dx.doi.org/10.1016/j.nuclphysa.2015.03.004

http://dx.doi.org/10.1126/science.1233232

http://dx.doi.org/10.1143/PTP.85.1287

http://dx.doi.org/10.1016/0370-2693(95)00688-H

http://dx.doi.org/10.1093/ptep/pts010


http://dx.doi.org/10.1051/0004-6361:20066983

http://dx.doi.org/10.1051/0004-6361:20066983

http://dx.doi.org/10.1088/0954-3899/40/1/013201

http://dx.doi.org/10.1088/0954-3899/40/1/013201










http://dx.doi.org/10.1142/S0218301307008884









http://dx.doi.org/10.1103/PhysRevE.64.057402


http://dx.doi.org/10.1016/S0375-9474(97)81465-0

http://dx.doi.org/10.1007/3-540-44578-1_1

http://dx.doi.org/10.1088/0034-4885/75/2/026301

http://dx.doi.org/10.1134/S1063778814080031




Baldo, M., A. Fiasconaro, H. Q. Song, G. Giansiracusa, and U.Lombardo, 2001, Phys. Rev. C 65, 017303.

Baldo, M., and C. Maieron, 2004, Phys. Rev. C 69, 014301.Baldo, M., C. Maieron, P. Schuck, and X. Viñas, 2004, Nucl. Phys. A736, 241.

Baldo, M., A. Polls, A. Rios, H.-J. Schulze, and I. Vidaña, 2012,Phys. Rev. C 86, 064001.

Baldo, M., L. Robledo, P. Schuck, and X. Viñas, 2013, Phys. Rev. C87, 064305.

Baldo, M., P. Schuck, and X. Viñas, 2008, Phys. Lett. B 663, 390.Banik, S., M. Hempel, and D. Bandyopadhyay, 2014, Astrophys. J.Suppl. Ser. 214, 22.

Baran, V., M. Colonna, V. Greco, and M. Di Toro, 2005, Phys. Rep.410, 335.

Barkat, Z., J.-R. Buchler, and L. Ingber, 1972, Astrophys. J. 176, 723.Barranco, M., and J. R. Buchler, 1980, Phys. Rev. C 22, 1729.Bart, S., et al., 1999, Phys. Rev. Lett. 83, 5238.Bartlett, R. J., and M. Musial, 2007, Rev. Mod. Phys. 79, 291.Bashir, A., et al., 2012, Commun. Theor. Phys. 58, 79.Bauböck, M., F. Özel, D. Psaltis, and S. M. Morsink, 2015,Astrophys. J. 799, 22.

Bauböck, M., D. Psaltis, and F. Özel, 2013, Astrophys. J. 766, 87.Baumgarte, T. W., S. A. Teukolsky, S. L. Shapiro, H. T. Janka, andW.Keil, 1996, Astrophys. J. 468, 823.

Bauswein, A., T. Baumgarte, and H. T. Janka, 2013, Phys. Rev. Lett.111, 131101.

Bauswein, A., S. Goriely, and H. T. Janka, 2013, Astrophys. J. 773,78.

Bauswein, A., H. Janka, K. Hebeler, and A. Schwenk, 2012, Phys.Rev. D 86, 063001.

Bauswein, A., H.-T. Janka, and R. Oechslin, 2010, Phys. Rev. D 82,084043.

Bauswein, A., and N. Stergioulas, 2015, Phys. Rev. D 91, 124056.Bauswein, A., N. Stergioulas, and H.-T. Janka, 2014, Phys. Rev. D90, 023002.

Bauswein, A., N. Stergioulas, and H.-T. Janka, 2016, Eur. Phys. J. A52, 56.

Baym, G., H. A. Bethe, and C. Pethick, 1971, Nucl. Phys. A 175,225.

Baym, G., and C. Pethick, 1975, Annu. Rev. Nucl. Part. Sci. 25, 27.Baym, G., and C. Pethick, 1979, Annu. Rev. Astron. Astrophys. 17,415.

Baym, G., C. Pethick, and P. Sutherland, 1971, Astrophys. J. 170,299.

Beane, S., W. Detmold, K. Orginos, and M. Savage, 2011, Prog. Part.Nucl. Phys. 66, 1.

Beane, S., et al., 2012, Phys. Rev. Lett. 109, 172001.Beane, S. R., et al. (NPLQCD Collaboration), 2007, Nucl. Phys. A794, 62.

Bedaque, P. F., and G. Rupak, 2003, Phys. Rev. B 67, 174513.Bedaque, P. F., and U. van Kolck, 2002, Annu. Rev. Nucl. Part. Sci.52, 339.

Bednarek, I., P. Haensel, J. Zdunik, M. Bejger, and R. Manka, 2012,Astron. Astrophys. 543, A157.

Beiner, M., H. Flocard, N. van Giai, and P. Quentin, 1975, Nucl.Phys. A 238, 29.

Beisitzer, T., R. Stiele, and J. Schaffner-Bielich, 2014, Phys. Rev. D90, 085001.

Bender, M., P.-H. Heenen, and P.-G. Reinhard, 2003, Rev. Mod.Phys. 75, 121.

Bender, M., et al., 2009, Phys. Rev. C 80, 064302.Benic, S., D. Blaschke, D. E. Alvarez-Castillo, T. Fischer, and S.Typel, 2015, Astron. Astrophys. 577, A40.

Bentz, W., and A.W. Thomas, 2001, Nucl. Phys. A 696, 138.Bertsch, G., and S. Das Gupta, 1988, Phys. Rep. 160, 189.Beth, E., and G. Uhlenbeck, 1936, Physica (Utrecht) 3, 729.Beth, E., and G. Uhlenbeck, 1937, Physica (Utrecht) 4, 915.Bethe, H. A., 1936, Phys. Rev. 50, 332.Bhattacharyya, A., I. N. Mishustin, and W. Greiner, 2010, J. Phys. G37, 025201.

Blaschke, D., M. Buballa, A. Radzhabov, and M. Volkov, 2008,Phys. At. Nucl. 71, 1981.

Blaschke, D., H. Grigorian, and D. Voskresensky, 2004, Astron.Astrophys. 424, 979.

Blaschke, D., H. Grigorian, D. Voskresensky, and F. Weber, 2012,Phys. Rev. C 85, 022802.

Blaschke, D., D. Zablocki, M. Buballa, A. Dubinin, and G. Röpke,2014, Ann. Phys. (Amsterdam) 348, 228.

Blättel, B., V. Koch, and U. Mosel, 1993, Rep. Prog. Phys. 56, 1.Blinnikov, S., I. Panov, M. Rudzsky, and K. Sumiyoshi, 2011,Astron. Astrophys. 535, A37.

Bodmer, A., Q. Usmani, and J. Carlson, 1984, Phys. Rev. C 29, 684.Bogdanov, S., 2013, Astrophys. J. 762, 96.Bogner, S. K., R. J. Furnstahl, and A. Schwenk, 2010, Prog. Part.Nucl. Phys. 65, 94.

Boguta, J., and A. Bodmer, 1977, Nucl. Phys. A 292, 413.Bombaci, I., A. Fabrocini, A. Polls, and I. Vidaña, 2005, Phys. Lett.B 609, 232.

Bombaci, I., D. Logoteta, C. Providência, and I. Vidaña, 2007,Astron. Astrophys. 462, 1017.

Bonanno, L., and A. Sedrakian, 2012, Astron. Astrophys. 539, A16.Bonche, P., S. Levit, and D. Vautherin, 1985, Nucl. Phys. A 436, 265.Bonche, P., and D. Vautherin, 1981, Nucl. Phys. A 372, 496.Bonche, P., and D. Vautherin, 1982, Astron. Astrophys. 112, 268.Bondorf, J. P., A. S. Botvina, A. S. Iljinov, I. N. Mishustin, and K.Sneppen, 1995, Phys. Rep. 257, 133.

Botvina, A. S., and I. N. Mishustin, 2004, Phys. Lett. B 584, 233.Botvina, A. S., and I. N. Mishustin, 2010, Nucl. Phys. A 843, 98.Bowler, R., and M. Birse, 1995, Nucl. Phys. A 582, 655.Bożek, P., and P. Czerski, 2001, Eur. Phys. J. A 11, 271.Brack, M., C. Guet, and H.-B. Hakansson, 1985, Phys. Rep. 123,275.

Brack, M., B. Jennings, and Y. Chu, 1976, Phys. Lett. B 65, 1.Brack, M., and P. Quentin, 1974a, Phys. Scr. 10A, 163.Brack, M., and P. Quentin, 1974b, Phys. Lett. B 52, 159.Brink, D., and E. Boeker, 1967, Nucl. Phys. A 91, 1.Brito, L., et al., 2006, Phys. Rev. C 74, 045801.Brockmann, R., and R. Machleidt, 1984, Phys. Lett. B 149, 283.Brockmann, R., and R. Machleidt, 1990, Phys. Rev. C 42, 1965.Brown, B. A., 2000, Phys. Rev. Lett. 85, 5296.Brown, B. A., G. Shen, G. Hillhouse, J. Meng, and A. Trzcinska,2007, Phys. Rev. C 76, 034305.

Brown, G. E., 1987, Phys. Scr. 36, 209.Brown, G. E., V. Thorsson, K. Kubodera, and M. Rho, 1992,Phys. Lett. B 291, 355.

Bruenn, S. W., 1985, Astrophys. J. Suppl. Ser. 58, 771.Brush, S., H. Sahlin, and E. Teller, 1966, J. Chem. Phys. 45, 2102.Buballa, M., 2005, Phys. Rep. 407, 205.Buballa, M., and M. Oertel, 1999, Phys. Lett. B 457, 261.Buballa, M., et al., 2014, J. Phys. G 41, 123001.Buchler, J.-R., and Z. Barkat, 1971, Phys. Rev. Lett. 27, 48.Buchler, J.-R., and S. A. Coon, 1977, Astrophys. J. 212, 807.Burgio, G. F., H.-J. Schulze, and A. Li, 2011, Phys. Rev. C 83,025804.

Burrows, A., 2013, Rev. Mod. Phys. 85, 245.Burrows, A., and J. M. Lattimer, 1984, Astrophys. J. 285, 294.











http://dx.doi.org/10.1088/0067-0049/214/2/22

http://dx.doi.org/10.1088/0067-0049/214/2/22

http://dx.doi.org/10.1016/j.physrep.2004.12.004


http://dx.doi.org/10.1086/151674




http://dx.doi.org/10.1088/0253-6102/58/1/16

http://dx.doi.org/10.1088/0004-637X/799/1/22

http://dx.doi.org/10.1088/0004-637X/766/2/87

http://dx.doi.org/10.1086/177738



http://dx.doi.org/10.1088/0004-637X/773/1/78

http://dx.doi.org/10.1088/0004-637X/773/1/78










http://dx.doi.org/10.1016/0375-9474(71)90281-8

http://dx.doi.org/10.1016/0375-9474(71)90281-8

http://dx.doi.org/10.1146/annurev.ns.25.120175.000331

http://dx.doi.org/10.1146/annurev.aa.17.090179.002215

http://dx.doi.org/10.1146/annurev.aa.17.090179.002215

http://dx.doi.org/10.1086/151216

http://dx.doi.org/10.1086/151216

http://dx.doi.org/10.1016/j.ppnp.2010.08.002





http://dx.doi.org/10.1103/PhysRevB.67.174513

http://dx.doi.org/10.1146/annurev.nucl.52.050102.090637


http://dx.doi.org/10.1051/0004-6361/201118560

http://dx.doi.org/10.1016/0375-9474(75)90338-3

http://dx.doi.org/10.1016/0375-9474(75)90338-3






http://dx.doi.org/10.1051/0004-6361/201425318

http://dx.doi.org/10.1016/S0375-9474(01)01119-8

http://dx.doi.org/10.1016/0370-1573(88)90170-6

http://dx.doi.org/10.1016/S0031-8914(36)80346-2

http://dx.doi.org/10.1016/S0031-8914(37)80189-5


http://dx.doi.org/10.1088/0954-3899/37/2/025201

http://dx.doi.org/10.1088/0954-3899/37/2/025201

http://dx.doi.org/10.1134/S1063778808110161

http://dx.doi.org/10.1051/0004-6361:20040404

http://dx.doi.org/10.1051/0004-6361:20040404


http://dx.doi.org/10.1016/j.aop.2014.06.002

http://dx.doi.org/10.1088/0034-4885/56/1/001

http://dx.doi.org/10.1051/0004-6361/201117225


http://dx.doi.org/10.1088/0004-637X/762/2/96



http://dx.doi.org/10.1016/0375-9474(77)90626-1



http://dx.doi.org/10.1051/0004-6361:20065259

http://dx.doi.org/10.1051/0004-6361/201117832

http://dx.doi.org/10.1016/0375-9474(85)90199-X

http://dx.doi.org/10.1016/0375-9474(81)90049-X

http://dx.doi.org/10.1016/0370-1573(94)00097-M



http://dx.doi.org/10.1016/0375-9474(94)00481-2

http://dx.doi.org/10.1007/s100500170064

http://dx.doi.org/10.1016/0370-1573(86)90078-5

http://dx.doi.org/10.1016/0370-1573(86)90078-5

http://dx.doi.org/10.1016/0370-2693(76)90519-0

http://dx.doi.org/10.1088/0031-8949/10/A/028

http://dx.doi.org/10.1016/0370-2693(74)90077-X

http://dx.doi.org/10.1016/0375-9474(67)90446-0


http://dx.doi.org/10.1016/0370-2693(84)90407-6




http://dx.doi.org/10.1088/0031-8949/36/2/003

http://dx.doi.org/10.1016/0370-2693(92)91386-N

http://dx.doi.org/10.1086/191056

http://dx.doi.org/10.1063/1.1727895


http://dx.doi.org/10.1016/S0370-2693(99)00533-X

http://dx.doi.org/10.1088/0954-3899/41/12/123001


http://dx.doi.org/10.1086/155106




http://dx.doi.org/10.1086/162505

Bürvenich, T., D. Madland, J. Maruhn, and P. Reinhard, 2002,Phys. Rev. C 65, 044308.

Buss, O., et al., 2012, Phys. Rep. 512, 1.Buyukcizmeci, N., A. S. Botvina, and I. N. Mishustin, 2014,Astrophys. J. 789, 33.

Buyukcizmeci, N., et al., 2013, Nucl. Phys. A 907, 13.Caballero, O., S. Postnikov, C. Horowitz, and M. Prakash, 2008,Phys. Rev. C 78, 045805.

Cahill, R., J. Praschifka, and C. Burden, 1989, Aust. J. Phys. 42, 161.Cahill, R., and C. D. Roberts, 1985, Phys. Rev. C 32, 2419.Caplan, M., A. Schneider, C. Horowitz, and D. Berry, 2014,arXiv:1012.3208.

Carbone, A., 2014, Self-consistent Green’s functions with three-bodyforces, Ph.D. thesis, University of Barcelona, arXiv:1407.6622.

Carbone, A., A. Cipollone, C. Barbieri, A. Rios, and A. Polls, 2013,Phys. Rev. C 88, 054326.

Carbone, A., G. Colo, A. Bracco, L.-G. Cao, P. F. Bortignon, F.Camera, and O. Wieland, 2010, Phys. Rev. C 81, 041301.

Carlson, J., 1987, Phys. Rev. C 36, 2026.Carlson, J., 1988, Phys. Rev. C 38, 1879.Carlson, J., S. Gandolfi, and A. Gezerlis, 2012, Prog. Theor. Exp.Phys. 2012, 01A209.

Carlson, J., S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E.Schmidt, and R. B. Wiringa, 2015, Rev. Mod. Phys. 87, 1067.

Carlson, J., J. Morales, Jr., V. Pandharipande, and D. Ravenhall,2003, Phys. Rev. C 68, 025802.

Carlson, J., V. Pandharipande, and R. B. Wiringa, 1983, Nucl. Phys.A 401, 59.

Carnahan, N. F., and K. E. Starling, 1969, J. Chem. Phys. 51,635.

Carter, B., N. Chamel, and P. Haensel, 2005a, Nucl. Phys. A 759,441.

Carter, B., N. Chamel, and P. Haensel, 2005b, Nucl. Phys. A 748,675.

Carter, B., N. Chamel, and P. Haensel, 2006, Int. J. Mod. Phys. D 15,777.

Centelles, M., X. Roca-Maza, X. Viñas, and M. Warda, 2009,Phys. Rev. Lett. 102, 122502.

Centelles, M., X. Roca-Maza, X. Viñas, and M. Warda, 2010, Phys.Rev. C 82, 054314.

Cerdá-Durán, P., N. DeBrye, M. A. Aloy, J. A. Font, and M.Obergaulinger, 2013, Astrophys. J. 779, L18.

Chabanat, E., P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer,1998, Nucl. Phys. A 635, 231.

Chabanat, E., J. Meyer, P. Bonche, R. Schaeffer, and P. Haensel,1997, Nucl. Phys. A 627, 710.

Chabrier, G., F. Douchin, and A. Potekhin, 2002, J. Phys. Condens.Matter 14, 9133.

Chabrier, G., and A. Y. Potekhin, 1998, Phys. Rev. E 58, 4941.Chajecki, Z., et al., 2014, arXiv:1402.5216.Chamel, N., 2005, Nucl. Phys. A 747, 109.Chamel, N., A. Fantina, J. Pearson, and S. Goriely, 2011, Phys. Rev.C 84, 062802.

Chamel, N., and S. Goriely, 2010, Phys. Rev. C 82, 045804.Chamel, N., and P. Haensel, 2006, Phys. Rev. C 73, 045802.Chamel, N., and P. Haensel, 2008, Living Rev. Relativ. 11, 10.Char, P., S. Banik, and D. Bandyopadhyay, 2015, Astrophys. J. 809,116.

Chatterjee, D., and I. Vidaña, 2016, Eur. Phys. J. A 52, 29.Chen, H., M. Baldo, G. Burgio, and H.-J. Schulze, 2011, Phys. Rev.D 84, 105023.

Chen, H., M. Baldo, G. F. Burgio, and H.-J. Schulze, 2012, Phys.Rev. D 86, 045006.

Chen, H., J.-B. Wei, M. Baldo, G. Burgio, and H.-J. Schulze, 2015,Phys. Rev. D 91, 105002.

Chen, H., et al., 2008, Phys. Rev. D 78, 116015.Chen, L.-W., 2011, Phys. Rev. C 83, 044308.Chen, L.-W., C. M. Ko, and B.-A. Li, 2005a, Phys. Rev. Lett. 94,032701.

Chen, L.-W., C. M. Ko, and B.-A. Li, 2005b, Phys. Rev. C 72,064309.

Chen, L.-W., C. M. Ko, B.-A. Li, C. Xu, and J. Xu, 2014, Eur. Phys.J. A 50, 29.

Chen, L.-W., C. M. Ko, B.-A. Li, and J. Xu, 2010, Phys. Rev. C 82,024321.

Chen, Y., 2012, Eur. Phys. J. A 48, 132.Chen, Y., 2014, Phys. Rev. C 89, 064306.Cheng, K., C. Yao, and Z. Dai, 1997, Phys. Rev. C 55, 2092.Chin, S., and J. Walecka, 1974, Phys. Lett. B 52, 24.Chodos, A., R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F.Weisskopf, 1974, Phys. Rev. D 9, 3471.

Chomaz, P., F. Gulminelli, C. Ducoin, P. Napolitani, and K.Hasnaoui, 2007, Phys. Rev. C 75, 065805.

Chugunov, A. I., and H. E. DeWitt, 2009, Contrib. Plasma Phys. 49,696.

Coester, F., 1958, Nucl. Phys. 7, 421.Coester, F., and H. Kümmel, 1960, Nucl. Phys. 17, 477.Collins, J. C., and M. J. Perry, 1975, Phys. Rev. Lett. 34, 1353.Colo, G., U. Garg, and H. Sagawa, 2014, Eur. Phys. J. A 50, 26.Colucci, G., and A. Sedrakian, 2013, Phys. Rev. C 87, 055806.Constantinou, C., B. Muccioli, M. Prakash, and J. M. Lattimer, 2014,Phys. Rev. C 89, 065802.

Constantinou, C., B. Muccioli, M. Prakash, and J. M. Lattimer, 2015,Phys. Rev. C 92, 025801.

Contrera, G., A. Grunfeld, and D. Blaschke, 2014, Phys. Part. Nucl.Lett. 11, 342.

Coon, S., and H. Han, 2001, Few-Body Syst. 30, 131.Coon, S. A., and W. Glöckle, 1981, Phys. Rev. C 23, 1790.Cooper, E. D., S. Hama, B. C. Clark, and R. L. Mercer, 1993,Phys. Rev. C 47, 297.

Cooper, L. N., 1956, Phys. Rev. 104, 1189.Cooper, R. L., and L. Bildsten, 2008, Phys. Rev. E 77, 056405.Cottam, J., F. Paerels, and M. Mendez, 2002, Nature (London)420, 51.

Cottam, J., F. Paerels, M. Méndez, L. Boirin, W. H. G. Lewin, E.Kuulkers, and J. M. Miller, 2008, Astrophys. J. 672, 504.

Cottingham, W., M. Lacombe, B. Loiseau, J. Richard, and R. VinhMau, 1973, Phys. Rev. D 8, 800.

Coupland, D., et al., 2016, Phys. Rev. C 94, 011601.Cozma, M., 2011, Phys. Lett. B 700, 139.Cozma, M., Y. Leifels, W. Trautmann, Q. Li, and P. Russotto, 2013,Phys. Rev. C 88, 044912.

Daligault, J., 2006, Phys. Rev. E 73, 056407.Danielewicz, P., 1984a, Ann. Phys. (N.Y.) 152, 239.Danielewicz, P., 1984b, Ann. Phys. (N.Y.) 152, 305.Danielewicz, P., 2003, Nucl. Phys. A 727, 233.Danielewicz, P., R. Lacey, and W. G. Lynch, 2002, Science 298,1592.

Danielewicz, P., and J. Lee, 2014, Nucl. Phys. A 922, 1.Danysz, M., and J. Pniewski, 1953a, Bull. Acad. Pols. Sci. 1, 42.Danysz, M., and J. Pniewski, 1953b, Philos. Mag. Ser. 5 44, 348.Daoutidis, I., and S. Goriely, 2011, Phys. Rev. C 84, 027301.Dasgupta, B., T. Fischer, S. Horiuchi, M. Liebendörfer, A. Mirizzi, I.Sagert, and J. Schaffner-Bielich, 2010, Phys. Rev. D 81, 103005.

Dashen, R., S.-K. Ma, and H. J. Bernstein, 1969, Phys. Rev. 187,345.





http://dx.doi.org/10.1088/0004-637X/789/1/33



http://dx.doi.org/10.1071/PH890161

http://dx.doi.org/10.1103/PhysRevB.32.2419

http://arXiv.org/abs/1012.3208










http://dx.doi.org/10.1016/0375-9474(83)90336-6

http://dx.doi.org/10.1016/0375-9474(83)90336-6

http://dx.doi.org/10.1063/1.1672048

http://dx.doi.org/10.1063/1.1672048





http://dx.doi.org/10.1142/S0218271806008504

http://dx.doi.org/10.1142/S0218271806008504




http://dx.doi.org/10.1088/2041-8205/779/2/L18

http://dx.doi.org/10.1016/S0375-9474(98)00180-8

http://dx.doi.org/10.1016/S0375-9474(97)00596-4

http://dx.doi.org/10.1088/0953-8984/14/40/307

http://dx.doi.org/10.1088/0953-8984/14/40/307








http://dx.doi.org/10.12942/lrr-2008-10

http://dx.doi.org/10.1088/0004-637X/809/2/116

http://dx.doi.org/10.1088/0004-637X/809/2/116




















http://dx.doi.org/10.1016/0370-2693(74)90708-4



http://dx.doi.org/10.1002/ctpp.200910079


http://dx.doi.org/10.1016/0029-5582(58)90280-3

http://dx.doi.org/10.1016/0029-5582(60)90140-1






http://dx.doi.org/10.1134/S1547477114040128

http://dx.doi.org/10.1134/S1547477114040128

http://dx.doi.org/10.1007/s006010170022







http://dx.doi.org/10.1086/524186






http://dx.doi.org/10.1016/0003-4916(84)90092-7

http://dx.doi.org/10.1016/0003-4916(84)90093-9





http://dx.doi.org/10.1080/14786440308520318





Davesne, D., et al., 2015, Phys. Rev. C 91, 064303.Day, B., 1967, Rev. Mod. Phys. 39, 719.Day, B. D., and J. G. Zabolitzky, 1981, Nucl. Phys. A 366, 221.Dean, D. J., K. Langanke, and J. M. Sampaio, 2002, Phys. Rev. C 66,045802.

de Carvalho, S., M. Rotondo, J. A. Rueda, and R. Ruffini, 2014,Phys. Rev. C 89, 015801.

Dechargé, J., and D. Gogny, 1980, Phys. Rev. C 21, 1568.De Filippo, E., and A. Pagano, 2014, Eur. Phys. J. A 50, 32.de Forcrand, P., and O. Philipsen, 2002, Nucl. Phys. B 642, 290.Demorest, P., T. Pennucci, S. Ransom, M. Roberts, and J. Hessels,2010, Nature (London) 467, 1081.

De Vries, H., C.W. De Jager, and C. De Vries, 1987, At. Data Nucl.Data Tables 36, 495.

DeWitt, H., W. Slattery, and G. Chabrier, 1996, Physica B(Amsterdam) 228, 21.

Dexheimer, V., and S. Schramm, 2008, Astrophys. J. 683, 943.Dexheimer, V., and S. Schramm, 2010, Phys. Rev. C 81, 045201.Dickhoff, W., and C. Barbieri, 2004, Prog. Part. Nucl. Phys. 52, 377.Dieperink, A., Y. Dewulf, D. Van Neck, M. Waroquier, and V. Rodin,2003, Phys. Rev. C 68, 064307.

Di Toro, M., et al., 2009, Prog. Part. Nucl. Phys. 62, 389.Đapo, H., B.-J. Schaefer, and J. Wambach, 2008, Eur. Phys. J. A 36,101.

Đapo, H., B.-J. Schaefer, and J. Wambach, 2010, Phys. Rev. C 81,035803.

Dobaczewski, J., B. Carlsson, and M. Kortelainen, 2010, J. Phys. G37, 075106.

Donati, P., P. M. Pizzochero, P. F. Bortignon, and R. A. Broglia,1994, Phys. Rev. Lett. 72, 2835.

Dong, J., H. Zhang, L. Wang, and W. Zuo, 2013, Phys. Rev. C 88,014302.

Dong, J., W. Zuo, and J. Gu, 2013, Phys. Rev. C 87, 014303.Dong, J., W. Zuo, J. Gu, and U. Lombardo, 2012, Phys. Rev. C 85,034308.

Donoghue, J. F., 2006, Phys. Lett. B 643, 165.Dorso, C., P. Giménez Molinelli, and J. López, 2012, Phys. Rev. C86, 055805.

Dorso, C., P. Giménez Molinelli, J. Lopez, and E. Ramirez-Homs,2012, arXiv:1208.4841.

Dorso, C., P. Giménez Molinelli, J. Nichols, and J. Lopez, 2012,arXiv:1211.5582.

Dorso, C., and J. Randrup, 1993, Phys. Lett. B 301, 328.Dorso, C. O., P. A. G. Molinelli, and J. A. Lopez, 2011,arXiv:1111.1258.

Douchin, F., and P. Haensel, 2000, Phys. Lett. B 485, 107.Douchin, F., and P. Haensel, 2001, Astron. Astrophys. 380, 151.Douchin, F., P. Haensel, and J. Meyer, 2000, Nucl. Phys. A 665, 419.Downum, C., T. Barnes, J. Stone, and E. Swanson, 2006, Phys. Lett.B 638, 455.

Drago, A., A. Lavagno, and G. Pagliara, 2014, Phys. Rev. D 89,043014.

Drago, A., A. Lavagno, G. Pagliara, and D. Pigato, 2014, Phys. Rev.C 90, 065809.

Dreizler, R., and E. Gross, 1990, Density Functional Theory: AnApproach to the Quantum Many-Body Problem (Springer,New York).

Drischler, C., V. Soma, and A. Schwenk, 2014, Phys. Rev. C 89,025806.

Ducoin, C., P. Chomaz, and F. Gulminelli, 2006, Nucl. Phys. A 771,68.

Ducoin, C., J. Margueron, and P. Chomaz, 2008, Nucl. Phys. A809, 30.

Ducoin, C., J. Margueron, C. Providência, and I. Vidaña, 2011,Phys. Rev. C 83, 045810.

Ducoin, C., C. Providência, A. M. Santos, L. Brito, and P. Chomaz,2008, Phys. Rev. C 78, 055801.

Duflo, J., and A. P. Zuker, 1995, Phys. Rev. C 52, R23.Dutra, M., et al., 2012, Phys. Rev. C 85, 035201.Dutra, M., et al., 2014, Phys. Rev. C 90, 055203.Ebert, D., T. Feldmann, and H. Reinhardt, 1996, Phys. Lett. B 388,154.

Eisele, F., H. Filthuth, W. Foehlisch, V. Hepp, and G. Zech, 1971,Phys. Lett. B 37, 204.

El Eid, M., and W. Hillebrandt, 1980, Astron. Astrophys. Suppl. Ser.42, 215.

Ellis, J. R., J. I. Kapusta, and K. A. Olive, 1991, Nucl. Phys. B 348,345.

Elshamouty, K. G., C. O. Heinke, G. R. Sivakoff, W. C. G. Ho, P. S.Shternin, D. G. Yakovlev, D. J. Patnaude, and L. David, 2013,Astrophys. J. 777, 22.

Endo, T., T. Maruyama, S. Chiba, and T. Tatsumi, 2006, Prog. Theor.Phys. 115, 337.

Engelbrecht, C. A., and J. R. Engelbrecht, 1991, Ann. Phys. (N.Y.)207, 1.

Engelmann, R., H. Filthuth, V. Hepp, and E. Kluge, 1966, Phys. Lett.21, 587.

Epelbaum, E., H.-W. Hammer, and U.-G. Meissner, 2009, Rev. Mod.Phys. 81, 1773.

Epelbaum, E., et al., 2014, Phys. Rev. Lett. 112, 102501.Erler, J., N. Birge, M. Kortelainen, W. Nazarewicz, E. Olsen, A. M.Perhac, and M. Stoitsov, 2012, Nature (London) 486, 509.

Erler, J., C. Horowitz, W. Nazarewicz, M. Rafalski, and P.-G.Reinhard, 2013, Phys. Rev. C 87, 044320.

Erler, J., P. Klüpfel, and P. G. Reinhard, 2010, Phys. Rev. C 82,044307.

Faber, J. A., and F. A. Rasio, 2012, Living Rev. Relativ. 15, 8.Fái, G., and J. Randrup, 1982, Nucl. Phys. A 381, 557.Falanga, M., E. Bozzo, A. Lutovinov, J. M. Bonnet-Bidaud, Y.Fetisova, and J. Puls, 2015, Astron. Astrophys. 577, A130.

Fan, Y.-Z., X.-F. Wu, and D.-M.Wei, 2013, Phys. Rev. D 88, 067304.Fantina, A., 2015 (private communication).Fantina, A. F., P. Blottiau, J. Margueron, P. Mellor, and P. M.Pizzochero, 2012, Astron. Astrophys. 541, A30.

Fantina, A. F., N. Chamel, J. M. Pearson, and S. Goriely, 2013,Astron. Astrophys. 559, A128.

Fantoni, S., and A. Fabrocini, 1998, inMicroscopic Quantum Many-Body Theories and their Applications, edited by J. Navarro and A.Polls, Vol. 510 of Lecture Notes in Physics (Springer-Verlag,Berlin).

Fantoni, S., and S. Rosati, 1975, Nuovo Cimento Soc. Ital. Fis., A 25,593.

Farhi, E., and R. Jaffe, 1984, Phys. Rev. D 30, 2379.Farouki, R., and S. Hamaguchi, 1993, Phys. Rev. E 47, 4330.Fattoyev, F. J., C. J. Horowitz, J. Piekarewicz, and G. Shen, 2010,Phys. Rev. C 82, 055803.

Fattoyev, F. J., and J. Piekarewicz, 2011, Phys. Rev. C 84, 064302.Fayans, S., 1998, J. Exp. Theor. Phys. Lett. 68, 169.Fayans, S., S. Tolokonnikov, E. Trykov, and D. Zawischa, 2000,Nucl. Phys. A 676, 49.

Fayans, S., and D. Zawischa, 2001, Int. J. Mod. Phys. B 15, 1684.Fedoseew, A., and H. Lenske, 2015, Phys. Rev. C 91, 034307.Ferdman, R. D., et al., 2014, Mon. Not. R. Astron. Soc. 443,2183.

Fetter, A., and J. Walecka, 1971, Quantum theory of many-particlesystems (McGraw-Hill, New York).





http://dx.doi.org/10.1016/0375-9474(81)90285-2





http://dx.doi.org/10.1140/epja/i2014-14032-y

http://dx.doi.org/10.1016/S0550-3213(02)00626-0


http://dx.doi.org/10.1016/0092-640X(87)90013-1

http://dx.doi.org/10.1016/0092-640X(87)90013-1

http://dx.doi.org/10.1016/S0921-4526(96)00330-4

http://dx.doi.org/10.1016/S0921-4526(96)00330-4

http://dx.doi.org/10.1086/589735









http://dx.doi.org/10.1088/0954-3899/37/7/075106

http://dx.doi.org/10.1088/0954-3899/37/7/075106












http://dx.doi.org/10.1016/0370-2693(93)91158-J


http://dx.doi.org/10.1016/S0370-2693(00)00672-9

http://dx.doi.org/10.1051/0004-6361:20011402

http://dx.doi.org/10.1016/S0375-9474(99)00397-8















http://dx.doi.org/10.1103/PhysRevC.52.R23



http://dx.doi.org/10.1016/0370-2693(96)01158-6

http://dx.doi.org/10.1016/0370-2693(96)01158-6

http://dx.doi.org/10.1016/0370-2693(71)90053-0

http://dx.doi.org/10.1016/0550-3213(91)90523-Z

http://dx.doi.org/10.1016/0550-3213(91)90523-Z

http://dx.doi.org/10.1088/0004-637X/777/1/22



http://dx.doi.org/10.1016/0003-4916(91)90176-9

http://dx.doi.org/10.1016/0003-4916(91)90176-9

http://dx.doi.org/10.1016/0031-9163(66)91310-2

http://dx.doi.org/10.1016/0031-9163(66)91310-2









http://dx.doi.org/10.1016/0375-9474(82)90376-1

http://dx.doi.org/10.1051/0004-6361/201425191


http://dx.doi.org/10.1051/0004-6361/201118187

http://dx.doi.org/10.1051/0004-6361/201321884

http://dx.doi.org/10.1007/BF02729302

http://dx.doi.org/10.1007/BF02729302





http://dx.doi.org/10.1134/1.567841

http://dx.doi.org/10.1016/S0375-9474(00)00192-5

http://dx.doi.org/10.1142/S0217979201006203


http://dx.doi.org/10.1093/mnras/stu1223


Feynman, R., N. Metropolis, and E. Teller, 1949, Phys. Rev. 75,1561.

Finelli, P., N. Kaiser, D. Vretenar, and W. Weise, 2003, Eur. Phys. J.A 17, 573.

Finelli, P., N. Kaiser, D. Vretenar, and W.Weise, 2004, Nucl. Phys. A735, 449.

Finelli, P., N. Kaiser, D. Vretenar, and W.Weise, 2006, Nucl. Phys. A770, 1.

Fiolhais, C., F. Nogueira, and M. Marques, 2003, Eds., A Primer inDensity Functional Theory, Vol. 620 of Lecture Notes in Physics(Springer, New York).

Fiorilla, S., N. Kaiser, and W. Weise, 2012, Nucl. Phys. A 880, 65.Fischer, C. S., J. Luecker, and J. A. Mueller, 2011, Phys. Lett. B 702,438.

Fischer, T., M. Hempel, I. Sagert, Y. Suwa, and J. Schaffner-Bielich,2014, Eur. Phys. J. A 50, 46.

Fischer, T., T. Klähn, I. Sagert, M. Hempel, and D. Blaschke, 2014,Acta Phys. Pol. B Proc. Suppl. 7, 153.

Fischer, T., I. Sagert, G. Pagliara, M. Hempel, J. Schaffner-Bielich, T.Rauscher, F.-K. Thielemann, R. Käppeli, G. Martínez-Pinedo, andM. Liebendörfer, 2011, Astrophys. J. Suppl. Ser. 194, 39.

Fischer, T., S. C. Whitehouse, A. Mezzacappa, F.-K. Thielemann,and M. Liebendörfer, 2009, Astron. Astrophys. 499, 1.

Fischer, T., S. C. Whitehouse, A. Mezzacappa, F.-K. Thielemann,and M. Liebendörfer, 2010, Astron. Astrophys. 517, A80.

Fischer, T., et al., 2012, Phys. At. Nucl. 75, 613.Fodor, Z., and S. Katz, 2002, Phys. Lett. B 534, 87.Fonseca, E., et al., 2016, Astrophys. J. 832, 167.Fortin, M., J. L. Zdunik, P. Haensel, and M. Bejger, 2015, Astron.Astrophys. 576, A68.

Foucart, F., R. Haas, M. D. Duez, E. O’Connor, C. D. Ott, L. Roberts,L. E. Kidder, J. Lippuner, H. P. Pfeiffer, and M. A. Scheel, 2016,Phys. Rev. D 93, 044019.

Fraga, E. S., R. D. Pisarski, and J. Schaffner-Bielich, 2001, Phys.Rev. D 63, 121702.

Freedman, B. A., and L. D. McLerran, 1977a, Phys. Rev. D 16, 1130.Freedman, B. A., and L. D. McLerran, 1977b, Phys. Rev. D 16, 1147.Freedman, B. A., and L. D. McLerran, 1977c, Phys. Rev. D 16, 1169.Friar, J. L., D. Huber, and U. van Kolck, 1999, Phys. Rev. C 59, 53.Frick, T., 2004, “Self-Consistent Green’s Functions in NuclearMatter at Finite Temperature,” Ph.D. thesis, Universität Tübingen.

Friedman, B., and V. Pandharipande, 1981, Nucl. Phys. A 361, 502.Friedman, J. L., and N. Stergioulas, 2013, Rotating Relativistic Stars(Cambridge University Press, Cambridge, UK).

Fritsch, S., N. Kaiser, and W. Weise, 2005, Nucl. Phys. A 750, 259.Fröhlich, C., G.Martínez-Pinedo,M. Liebendörfer, F. K. Thielemann,E. Bravo, W. R. Hix, K. Langanke, and N. T. Zinner, 2006,Phys. Rev. Lett. 96, 142502.

Fröhlich, C., et al., 2006, Astrophys. J. 637, 415.Fryer, C. L., K. Belczynski, E. Ramirez-Ruiz, S. Rosswog, G. Shen,and A.W. Steiner, 2015, Astrophys. J. 812, 24.

Fuchs, C., P. Essler, T. Gaitanos, and H. H. Wolter, 1997, Nucl. Phys.A 626, 987.

Fuchs, C., A. Faessler, E. Zabrodin, and Y.-M. Zheng, 2001,Phys. Rev. Lett. 86, 1974.

Fuchs, C., H. Lenske, and H. Wolter, 1995, Phys. Rev. C 52, 3043.Fuchs, C., and H. Wolter, 2006, Eur. Phys. J. A 30, 5.Fujita, J., and H. Miyazawa, 1957, Prog. Theor. Phys. 17, 360.Fukuda, T., et al. (E224 Collaboration), 1998, Phys. Rev. C 58, 1306.Fukushima, K., 2004, Phys. Lett. B 591, 277.Fukushima, K., 2008, Phys. Rev. D 77, 114028.Furnstahl, R., 2002, Nucl. Phys. A 706, 85.Furnstahl, R., 2004, Lect. Notes Phys. 641, 1.

Furnstahl, R., and K. Hebeler, 2013, Rep. Prog. Phys. 76, 126301.Furnstahl, R., B. D. Serot, and H.-B. Tang, 1997, Nucl. Phys. A 615,441.

Furusawa, S., H. Nagakura, K. Sumiyoshi, and S. Yamada, 2013,Astrophys. J. 774, 78.

Furusawa, S., K. Sumiyoshi, S. Yamada, and H. Suzuki, 2013,Astrophys. J. 772, 95.

Furusawa, S., S. Yamada, K. Sumiyoshi, and H. Suzuki, 2011,Astrophys. J. 738, 178.

Gaidarov, M., A. Antonov, P. Sarriguren, and E. Moya de Guerra,2012, Phys. Rev. C 85, 064319.

Gaidarov, M., A. Antonov, P. Sarriguren, and E. Moya de Guerra,2014, AIP Conf. Proc. 1606, 180.

Gaitanos, T., and M. Kaskulov, 2012, Nucl. Phys. A 878, 49.Gaitanos, T., and M. Kaskulov, 2015, Nucl. Phys. A 940, 181.Gaitanos, T., M. Kaskulov, and U. Mosel, 2009, Nucl. Phys. A828, 9.

Gaitanos, T., and M.M. Kaskulov, 2013, Nucl. Phys. A 899, 133.Gal, A., E. V. Hungerford, and D. J. Millener, 2016, Rev. Mod. Phys.88, 035004.

Gale, C., S. Jeon, and B. Schenke, 2013, Int. J. Mod. Phys. A 28,1340011.

Gale, C., G. M. Welke, M. Prakash, S. J. Lee, and S. Das Gupta,1990, Phys. Rev. C 41, 1545.

Galloway, D. K., and N. Lampe, 2012, Astrophys. J. 747, 75.Gandolfi, S., J. Carlson, and S. Reddy, 2012, Phys. Rev. C 85, 032801.Gandolfi, S., A. Gezerlis, and J. Carlson, 2015, Annu. Rev. Nucl.Part. Sci. 65, 303.

Gandolfi, S., A. Y. Illarionov, K. Schmidt, F. Pederiva, and S.Fantoni, 2009, Phys. Rev. C 79, 054005.

Gearheart, M., W. G. Newton, J. Hooker, and B.-A. Li, 2011,Mon. Not. R. Astron. Soc. 418, 2343.

Geng, L., H. Toki, and J. Meng, 2005, Prog. Theor. Phys. 113, 785.Gentile, N. A., M. B. Aufderheide, G. J. Mathews, F. D. Swesty, andG. M. Fuller, 1993, Astrophys. J. 414, 701.

Gezerlis, A., et al., 2013, Phys. Rev. Lett. 111, 032501.Gezerlis, A., et al., 2014, Phys. Rev. C 90, 054323.Giménez Molinelli, P. A., and C. O. Dorso, 2015, Nucl. Phys. A 933,306.

Giménez Molinelli, P., J. Nichols, J. López, and C. Dorso, 2014,Nucl. Phys. A 923, 31.

Glendenning, N. K., 1982, Phys. Lett. B 114, 392.Glendenning, N. K., 1992, Phys. Rev. D 46, 1274.Glendenning, N. K., 1997, Compact stars: Nuclear physics, particlephysics, and general relativity (Springer, New York).

Glendenning, N. K., and S. A. Moszkowski, 1991, Phys. Rev. Lett.67, 2414.

Gögelein, P., and H. Müther, 2007, Phys. Rev. C 76, 024312.Gögelein, P., E. van Dalen, C. Fuchs, and H. Müther, 2008,Phys. Rev. C 77, 025802.

Gomes, R. O., V. Dexheimer, S. Schramm, and C. A. Z. Vasconcellos,2015, Astrophys. J. 808, 8.

Gorenstein, M. I., V. K. Petrov, and G. M. Zinovjev, 1981, Phys. Lett.B 106, 327.

Goriely, S., N. Chamel, and J. Pearson, 2009a, Eur. Phys. J. A 42, 547.Goriely, S., N. Chamel, and J. Pearson, 2009b, Phys. Rev. Lett. 102,152503.

Goriely, S., N. Chamel, and J. Pearson, 2013a, Phys. Rev. C 88,024308.

Goriely, S., N. Chamel, and J. Pearson, 2013b, Phys. Rev. C 88,061302.

Goriely, S., S. Hilaire, and A. J. Koning, 2008, Phys. Rev. C 78,064307.















http://dx.doi.org/10.5506/APhysPolBSupp.7.153

http://dx.doi.org/10.1088/0067-0049/194/2/39

http://dx.doi.org/10.1051/0004-6361/200811055

http://dx.doi.org/10.1051/0004-6361/200913106

http://dx.doi.org/10.1134/S1063778812050067

http://dx.doi.org/10.1016/S0370-2693(02)01583-6

http://dx.doi.org/10.3847/0004-637X/832/2/167

http://dx.doi.org/10.1051/0004-6361/201424800

http://dx.doi.org/10.1051/0004-6361/201424800








http://dx.doi.org/10.1016/0375-9474(81)90649-7



http://dx.doi.org/10.1086/498224

http://dx.doi.org/10.1088/0004-637X/812/1/24

http://dx.doi.org/10.1016/S0375-9474(97)00467-3

http://dx.doi.org/10.1016/S0375-9474(97)00467-3








http://dx.doi.org/10.1016/S0375-9474(02)00867-9

http://dx.doi.org/10.1007/978-3-540-39911-7_1

http://dx.doi.org/10.1088/0034-4885/76/12/126301

http://dx.doi.org/10.1016/S0375-9474(96)00472-1

http://dx.doi.org/10.1016/S0375-9474(96)00472-1

http://dx.doi.org/10.1088/0004-637X/774/1/78

http://dx.doi.org/10.1088/0004-637X/772/2/95

http://dx.doi.org/10.1088/0004-637X/738/2/178









http://dx.doi.org/10.1142/S0217751X13400113

http://dx.doi.org/10.1142/S0217751X13400113


http://dx.doi.org/10.1088/0004-637X/747/1/75


http://dx.doi.org/10.1146/annurev-nucl-102014-021957



http://dx.doi.org/10.1111/j.1365-2966.2011.19628.x


http://dx.doi.org/10.1086/173116






http://dx.doi.org/10.1016/0370-2693(82)90078-8






http://dx.doi.org/10.1088/0004-637X/808/1/8

http://dx.doi.org/10.1016/0370-2693(81)90546-3

http://dx.doi.org/10.1016/0370-2693(81)90546-3










Gourgoulhon, E., P. Grandclément, J.-A. Marck, and J. Novak, andK. Taniguchi, 2016, Astrophysics Source Code Library, recordascl:1608.018.

Greiner, W., and J. Maruhn, 1996, Nuclear Models (Springer-Verlag,Berlin, Heidelberg).

Grill, F., H. Pais, C. Providência, I. Vidaña, and S. S. Avancini, 2014,Phys. Rev. C 90, 045803.

Grill, F., C. Providência, and S. S. Avancini, 2012, Phys. Rev. C 85,055808.

Gross, D. H. E., 1990, Rep. Prog. Phys. 53, 605.Gross, D. J., and F. Wilczek, 1973, Phys. Rev. Lett. 30, 1343.Gross-Boelting, T., C. Fuchs, and A. Faessler, 1999, Nucl. Phys. A648, 105.

Grossjean, M. K., and H. Feldmeier, 1985, Nucl. Phys. 444, 113.Guardiola, R., 1998, in Microscopic Quantum Many-Body Theoriesand their Applications, edited by J. Navarro and A. Polls, Vol. 510of Lecture Notes in Physics (Springer-Verlag, Berlin).

Guillot, S., and R. E. Rutledge, 2014, Astrophys. J. 796, L3.Guillot, S., M. Servillat, N. A. Webb, and R. E. Rutledge, 2013,Astrophys. J. 772, 7.

Gulminelli, F., P. Chomaz, A. Raduta, and A. Raduta, 2003,Phys. Rev. Lett. 91, 202701.

Gulminelli, F., A. Raduta, and M. Oertel, 2012, Phys. Rev. C 86,025805.

Gulminelli, F., A. Raduta, M. Oertel, and J. Margueron, 2013,Phys. Rev. C 87, 055809.

Gulminelli, F., and A. R. Raduta, 2015, Phys. Rev. C 92, 055803.Gupta, N., and P. Arumugam, 2013, Phys. Rev. C 87, 028801.Gusakov, M. E., and P. Haensel, 2005, Nucl. Phys. A 761, 333.Gusakov, M. E., E. M. Kantor, and P. Haensel, 2009a, Phys. Rev. C79, 055806.

Gusakov, M. E., E. M. Kantor, and P. Haensel, 2009b, Phys. Rev. C80, 015803.

Gutierrez, E., A. Ahmad, A. Ayala, A. Bashir, and A. Raya, 2014,J. Phys. G 41, 075002.

Güver, T., and F. Özel, 2013, Astrophys. J. 765, L1.Haensel, P., and B. Pichon, 1994, Astron. Astrophys. 283, 313.Haensel, P., A. Potekhin, and D. Yakovlev, 2007, Neutron stars 1:Equation of state and structure (Springer, New York).

Haensel, P., and J. Zdunik, 1990, Astron. Astrophys. 229, 117.Haensel, P., J. Zdunik, M. Bejger, and J. Lattimer, 2009, Astron.Astrophys. 502, 605.

Haensel, P., J. Zdunik, and J. Dobaczewski, 1989, Astron. Astrophys.222, 353.

Haensel, P., J. Zdunik, and R. Schaeffer, 1986, Astron. Astrophys.160, 121.

Hagel, K., et al., 2012, Phys. Rev. Lett. 108, 062702.Hagen, G., M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, and T.Papenbrock, 2012, Phys. Rev. Lett. 109, 032502.

Hagen, G., T. Papenbrock, D. J. Dean, and M. Hjorth-Jensen, 2010,Phys. Rev. C 82, 034330.

Hagen, G., T. Papenbrock, D. J. Dean, A. Schwenk, A. Nogga, M.Wloch, and P. Piecuch, 2007, Phys. Rev. C 76, 034302.

Hagen, G., et al., 2014, Phys. Rev. C 89, 014319.Haidenbauer, J., and U.-G. Meissner, 2005, Phys. Rev. C 72,044005.

Haidenbauer, J., U.-G. Meißner, A. Nogga, and H. Polinder, 2007,Lect. Notes Phys. 724, 113.

Haidenbauer, J., et al., 2013, Nucl. Phys. A 915, 24.Hama, S., B. C. Clark, E. D. Cooper, H. S. Sherif, and R. L. Mercer,1990, Phys. Rev. C 41, 2737.

Hansen, J. P., 1973, Phys. Rev. A 8, 3096.Hartle, J. B., and K. S. Thorne, 1968, Astrophys. J. 153, 807.

Hartnack, C., H. Oeschler, and J. Aichelin, 2006, Phys. Rev. Lett. 96,012302.

Hartnack, C., H. Oeschler, Y. Leifels, E. L. Bratkovskaya, and J.Aichelin, 2012, Phys. Rep. 510, 119.

Hartnack, C., et al., 1998, Eur. Phys. J. A 1, 151.Hashimoto, M.-a., H. Seki, and M. Yamada, 1984, Prog. Theor. Phys.71, 320.

Hashimoto, O., and H. Tamura, 2006, Prog. Part. Nucl. Phys. 57, 564.Hashimoto, T., et al., 2015, Phys. Rev. C 92, 031305.Hasnaoui, K., and J. Piekarewicz, 2013, Phys. Rev. C 88, 025807.Hatsuda, T., and T. Kunihiro, 1994, Phys. Rep. 247, 221.Hebeler, K., J. Lattimer, C. Pethick, and A. Schwenk, 2013,Astrophys. J. 773, 11.

Hebeler, K., and A. Schwenk, 2010, Phys. Rev. C 82, 014314.Hebeler, K., and A. Schwenk, 2014, Eur. Phys. J. A 50, 11.Heckel, S., P. P. Schneider, and A. Sedrakian, 2009, Phys. Rev. C 80,015805.

Heinke, C., et al., 2014, Mon. Not. R. Astron. Soc. 444, 443.Heinke, C. O., and W. C. Ho, 2010, Astrophys. J. 719, L167.Heiselberg, H., and V. Pandharipande, 2000, Annu. Rev. Nucl. Part.Sci. 50, 481.

Heiselberg, H., C. J. Pethick, and D. G. Ravenhall, 1993, Ann. Phys.(N.Y.) 223, 37.

Heiselberg, H., C. J. Pethick, and E. F. Staubo, 1993, Phys. Rev. Lett.70, 1355.

Hellemans, V., P. Heenen, and M. Bender, 2012, Phys. Rev. C 85,014326.

Hellemans, V., et al., 2013, Phys. Rev. C 88, 064323.Hempel, M., 2011, “Equations of State,” http://phys‑merger.physik.unibas.ch/~hempel/eos.html.

Hempel, M., V. Dexheimer, S. Schramm, and I. Iosilevskiy, 2013,Phys. Rev. C 88, 014906.

Hempel, M., T. Fischer, J. Schaffner-Bielich, and M. Liebendörfer,2012, Astrophys. J. 748, 70.

Hempel, M., K. Hagel, J. Natowitz, G. Röpke, and S. Typel, 2015,Phys. Rev. C 91, 045805.

Hempel, M., G. Pagliara, and J. Schaffner-Bielich, 2009, Phys. Rev.D 80, 125014.

Hempel, M., and J. Schaffner-Bielich, 2010, Nucl. Phys. A 837, 210.Hempel, M., J. Schaffner-Bielich, S. Typel, and G. Röpke, 2011,Phys. Rev. C 84, 055804.

Hessels, J. W., et al., 2006, Science 311, 1901.Hillebrandt, W., K. Nomoto, and R. G. Wolff, 1984, Astron.Astrophys. 133, 175.

Hillebrandt, W., and R. G. Wolff, 1985, “Models of Type II Super-nova Explosions,” Nucleosynthesis: Challenges and New Develop-ments, edited by W. D. Arnett and J. W. Truran (University ofChicago Press, Chicago), p. 131.

Hix, W. R., O. E. B. Messer, A. Mezzacappa, M. Liebendoerfer, J.Sampaio, K. Langanke, D. J. Dean, and G. Martínez-Pinedo, 2003,Phys. Rev. Lett. 91, 201102.

Ho, T.-L., and E. J. Mueller, 2004, Phys. Rev. Lett. 92, 160404.Ho, T.-L., and N. Zahariev, 2004, arXiv:cond-mat/0408469.Hofmann, F., C. Keil, and H. Lenske, 2001a, Phys. Rev. C 64,025804.

Hofmann, F., C. Keil, andH. Lenske, 2001b, Phys. Rev. C 64, 034314.Hohenberg, P., and W. Kohn, 1964, Phys. Rev. 136, B864.Holzenkamp, B., K. Holinde, and J. Speth, 1989, Nucl. Phys. A 500,485.

Horowitz, C., and D. Berry, 2008, Phys. Rev. C 78, 035806.Horowitz, C., and D. Berry, 2009, Phys. Rev. C 79, 065803.Horowitz, C., J. Hughto, A. Schneider, and D. Berry, 2011,arXiv:1109.5095.






http://dx.doi.org/10.1088/0034-4885/53/5/003


http://dx.doi.org/10.1016/S0375-9474(99)00022-6

http://dx.doi.org/10.1016/S0375-9474(99)00022-6

http://dx.doi.org/10.1016/0375-9474(85)90294-5

http://dx.doi.org/10.1088/2041-8205/796/1/L3

http://dx.doi.org/10.1088/0004-637X/772/1/7












http://dx.doi.org/10.1088/0954-3899/41/7/075002

http://dx.doi.org/10.1088/2041-8205/765/1/L1

http://dx.doi.org/10.1051/0004-6361/200811605

http://dx.doi.org/10.1051/0004-6361/200811605








http://dx.doi.org/10.1007/978-3-540-72039-3_4



http://dx.doi.org/10.1103/PhysRevA.8.3096

http://dx.doi.org/10.1086/149707




http://dx.doi.org/10.1007/s100500050045






http://dx.doi.org/10.1016/0370-1573(94)90022-1

http://dx.doi.org/10.1088/0004-637X/773/1/11






http://dx.doi.org/10.1088/2041-8205/719/2/L167



http://dx.doi.org/10.1006/aphy.1993.1026







http://phys-merger.physik.unibas.ch/%7Ehempel/eos.html






http://dx.doi.org/10.1088/0004-637X/748/1/70









http://arXiv.org/abs/cond-mat/0408469




http://dx.doi.org/10.1103/PhysRev.136.B864

http://dx.doi.org/10.1016/0375-9474(89)90223-6

http://dx.doi.org/10.1016/0375-9474(89)90223-6




Horowitz, C., M. Perez-Garcia, D. Berry, and J. Piekarewicz, 2005,Phys. Rev. C 72, 035801.

Horowitz, C., and J. Piekarewicz, 2002, Phys. Rev. C 66, 055803.Horowitz, C., S. Pollock, P. Souder, and R. Michaels, 2001,Phys. Rev. C 63, 025501.

Horowitz, C., and A. Schwenk, 2006a, Nucl. Phys. A 776, 55.Horowitz, C., and A. Schwenk, 2006b, Phys. Lett. B 642, 326.Horowitz, C., and A. Schwenk, 2006c, Phys. Lett. B 638, 153.Horowitz, C., and B. D. Serot, 1987, Nucl. Phys. A 464, 613.Horowitz, C., et al., 2012, Phys. Rev. C 85, 032501.Horowitz, C., et al., 2014, J. Phys. G 41, 093001.Horowitz, C., et al., 2015, Phys. Rev. Lett. 114, 031102.Horowitz, C. J., D. Berry, and E. Brown, 2007, Phys. Rev. E 75,066101.

Horowitz, C. J., M. Perez-Garcia, J. Carriere, D. Berry, and J.Piekarewicz, 2004, Phys. Rev. C 70, 065806.

Horowitz, C. J., M. Perez-Garcia, and J. Piekarewicz, 2004,Phys. Rev. C 69, 045804.

Horowitz, C. J., and J. Piekarewicz, 2001a, Phys. Rev. Lett. 86, 5647.Horowitz, C. J., and J. Piekarewicz, 2001b, Phys. Rev. C 64,062802.

Hotokezaka, K., K. Kiuchi, K. Kyutoku, T. Muranushi, Y.-i.Sekiguchi, M. Shibata, and K. Taniguchi, 2013, Phys. Rev. D88, 044026.

Hotokezaka, K., K. Kyutoku, M. Tanaka, K. Kiuchi, Y. Sekiguchi, M.Shibata, and S. Wanajo, 2013, Astrophys. J. Lett. 778, L16.

Huber, H., F. Weber, and M. Weigel, 1998, Phys. Rev. C 57, 3484.Hüdepohl, L., B. Müller, H.-T. Janka, A. Marek, and G. G. Raffelt,2010, Phys. Rev. Lett. 104, 251101.

Hugenholtz, N., and L. van Hove, 1958, Physica (Utrecht) 24, 363.Huizenga, J. R., and L. G. Moretto, 1972, Annu. Rev. Nucl. Part. Sci.22, 427.

Huovinen, P., and P. Ruuskanen, 2006, Annu. Rev. Nucl. Part. Sci.56, 163.

Ichimaru, S., H. Iyetomi, and S. Mitake, 1983, Astrophys. J. 265,L83.

Iliadis, C., 2007, Nuclear Physics of Stars (Wiley-VCH, New York).Iljinov, A. S., M. V. Mebel, N. Bianchi, E. De Sanctis, C. Guaraldo,V. Lucherini, V. Muccifora, E. Polli, A. R. Reolon, and P. Rossi,1992, Nucl. Phys. A 543, 517.

Inoue, T., et al. (HAL QCD Collaboration), 2010, Prog. Theor. Phys.124, 591.

Iosilevskiy, I., 2010, Acta Phys. Polonica B (Proc. Suppl.) 3, 589.Ishizuka, C., A. Ohnishi, and K. Sumiyoshi, 2003, Nucl. Phys. A723, 517.

Ishizuka, C., A. Ohnishi, K. Tsubakihara, K. Sumiyoshi, and S.Yamada, 2008, J. Phys. G 35, 085201.

Ishizuka, C., T. Suda, H. Suzuki, A. Ohnishi, K. Sumiyoshi, and H.Toki, 2015, Publ. Astron. Soc. Jpn. 67, 13.

Itoh, N., 1970, Prog. Theor. Phys. 44, 291.Jaikumar, P., S. Reddy, and A.W. Steiner, 2006, Phys. Rev. Lett. 96,041101.

Janka, H.-T., 2012a, Annu. Rev. Nucl. Part. Sci. 62, 407.Janka, H.-T., 2012b, arXiv:1206.2503.Janka, H.-T., K. Langanke, A. Marek, G. Martínez-Pinedo, and B.Müller, 2007, Phys. Rep. 442, 38.

Jastrzebski, J., et al., 2004, Int. J. Mod. Phys. E 13, 343.Jin, Z.-P., K. Hotokezaka, X. Li, M. Tanaka, P. D’Avanzo, Y.-Z.Fan, S. Covino, D.-M. Wei, and T. Piran, 2016, Nat. Commun. 7,12898.

Jog, C. J., and R. A. Smith, 1982, Astrophys. J. 253, 839.Juodagalvis, A., K. Langanke, W. R. Hix, G. Martínez-Pinedo, andJ. M. Sampaio, 2010, Nucl. Phys. A 848, 454.

Kaaret, P., Z. Prieskorn, J. J. M. in ’t Zand, S. Brandt, N. Lund, S.Mereghetti, D. Götz, E. Kuulkers, and J. A. Tomsick, 2007,Astrophys. J. Lett. 657, L97.

Kaiser, N., S. Fritsch, and W. Weise, 2002, Nucl. Phys. A 697,255.

Kaiser, N., S. Fritsch, and W. Weise, 2003, Nucl. Phys. A 724, 47.Kalantar-Nayestanaki, N., E. Epelbaum, J. Messchendorp, and A.Nogga, 2012, Rep. Prog. Phys. 75, 016301.

Kaplan, D. L., et al., 2013, Astrophys. J. 765, 158.Kastaun, W., and F. Galeazzi, 2015, Phys. Rev. D 91, 064027.Katayama, T., and K. Saito, 2014, arXiv:1410.7166.Kawaguchi, K., K. Kyutoku, M. Shibata, and M. Tanaka, 2016,Astrophys. J. 825, 52.

Keil, W., and H. T. Janka, 1995, Astron. Astrophys. 296, 145.Khan, E., and J. Margueron, 2013, arXiv:1304.4721.Khaustov, P., et al. (AGS E885 Collaboration), 2000, Phys. Rev. C61, 054603.

Kido, T., T. Maruyama, K. Niita, and S. Chiba, 2000, Nucl. Phys. A663–664, 877c.

Kitaura, F., H.-T. Janka, and W. Hillebrandt, 2006, Astron.Astrophys. 450, 345.

Klähn, T., and T. Fischer, 2015, Astrophys. J. 810, 134.Klähn, T., D. Blaschke, F. Sandin, C. Fuchs, A. Faessler, H.Grigorian, G. Röpke, and J. Trumper, 2007, Phys. Lett. B 654, 170.

Klähn, T., M. Oertel, and S. Typel, 2013, “CompOSE: CompstarOnline Supernova Equations of State,” http://compose.obspm.fr.

Klähn, T., C. D. Roberts, L. Chang, H. Chen, and Y.-X. Liu, 2010,Phys. Rev. C 82, 035801.

Klähn, T., et al., 2006, Phys. Rev. C 74, 035802.Kleinert, H., 1976, Phys. Lett. B 62, 429.Klevansky, S., 1992, Rev. Mod. Phys. 64, 649.Klimkiewicz, A., et al., 2007, Phys. Rev. C 76, 051603.Klos, B., et al., 2007, Phys. Rev. C 76, 014311.Klüpfel, P., P. G. Reinhard, T. J. Bürvenich, and J. A. Maruhn, 2009,Phys. Rev. C 79, 034310.

Kobyakov, D., and C. Pethick, 2014, Phys. Rev. Lett. 112, 112504.Köhler, H. S., 1976, Nucl. Phys. A 258, 301.Kohley, Z., and S. Yennello, 2014, Eur. Phys. J. A 50, 31.Kohley, Z., et al., 2010, Phys. Rev. C 82, 064601.Kohn, W., 1999, Rev. Mod. Phys. 71, 1253.Kohn, W., and L. Sham, 1965, Phys. Rev. 140, A1133.Kohno, M., Y. Fujiwara, Y. Watanabe, K. Ogata, and M. Kawai,2006, Phys. Rev. C 74, 064613.

Kojo, T., P. D. Powell, Y. Song, and G. Baym, 2015, Phys. Rev. D 91,045003.

Kong, H.-Y., Y. Xia, J. Xu, L.-W. Chen, B.-A. Li, and Y.-G. Ma,2015, Phys. Rev. C 91, 047601.

Kortelainen, M., and T. Lesinski, 2010, J. Phys. G 37, 064039.Kortelainen, M., et al., 2010, Phys. Rev. C 82, 024313.Kortelainen, M., et al., 2012, Phys. Rev. C 85, 024304.Kortelainen, M., et al., 2014, Phys. Rev. C 89, 054314.Kotake, K., K. Sato, and K. Takahashi, 2006, Rep. Prog. Phys. 69,971.

Koura, H., T. Tachibana, M. Uno, and M. Yamada, 2005, Prog.Theor. Phys. 113, 305.

Kowalski, S., et al., 2007, Phys. Rev. C 75, 014601.Krastev, P. G., and F. Sammarruca, 2006, Phys. Rev. C 74, 025808.Krasznahorkay, A., A. Balanda, J. Bordewijk, S. Brandenburg, M.Harakeh, N. Kalantar-Nayestanaki, B. Nyako, J. Timar, and A. vander Woude, 1994, Nucl. Phys. A 567, 521.

Krasznahorkay, A., et al., 1999, Phys. Rev. Lett. 82, 3216.Kreim, S., M. Hempel, D. Lunney, and J. Schaffner-Bielich, 2013,Int. J. Mass Spectrom. 349–350, 63.









http://dx.doi.org/10.1016/0375-9474(87)90370-8


http://dx.doi.org/10.1088/0954-3899/41/9/093001











http://dx.doi.org/10.1088/2041-8205/778/1/L16



http://dx.doi.org/10.1016/S0031-8914(58)95281-9





http://dx.doi.org/10.1086/183963

http://dx.doi.org/10.1086/183963

http://dx.doi.org/10.1016/0375-9474(92)90278-R



http://dx.doi.org/10.1016/S0375-9474(03)01324-1

http://dx.doi.org/10.1016/S0375-9474(03)01324-1

http://dx.doi.org/10.1088/0954-3899/35/8/085201

http://dx.doi.org/10.1093/pasj/psu141







http://dx.doi.org/10.1142/S0218301304002168

http://dx.doi.org/10.1038/ncomms12898


http://dx.doi.org/10.1086/159685


http://dx.doi.org/10.1086/513270

http://dx.doi.org/10.1016/S0375-9474(01)01231-3

http://dx.doi.org/10.1016/S0375-9474(01)01231-3

http://dx.doi.org/10.1016/S0375-9474(03)01475-1

http://dx.doi.org/10.1088/0034-4885/75/1/016301

http://dx.doi.org/10.1088/0004-637X/765/2/158



http://dx.doi.org/10.3847/0004-637X/825/1/52




http://dx.doi.org/10.1016/S0375-9474(99)00736-8

http://dx.doi.org/10.1016/S0375-9474(99)00736-8

http://dx.doi.org/10.1051/0004-6361:20054703

http://dx.doi.org/10.1051/0004-6361:20054703

http://dx.doi.org/10.1088/0004-637X/810/2/134


http://compose.obspm.fr





http://dx.doi.org/10.1016/0370-2693(76)90676-6






http://dx.doi.org/10.1016/0375-9474(76)90008-7




http://dx.doi.org/10.1103/PhysRev.140.A1133





http://dx.doi.org/10.1088/0954-3899/37/6/064039




http://dx.doi.org/10.1088/0034-4885/69/4/R03

http://dx.doi.org/10.1088/0034-4885/69/4/R03





http://dx.doi.org/10.1016/0375-9474(94)90022-1


http://dx.doi.org/10.1016/j.ijms.2013.02.015

Krüger, T., I. Tews, K. Hebeler, and A. Schwenk, 2013, Phys. Rev. C88, 025802.

Kubis, S., J. Porebska, and D. E. Alvarez-Castillo, 2010, Acta Phys.Pol. B 41, 2449.

Kümmel, H., K. H. Lührmann, and J. G. Zabolitzky, 1978, Phys.Rep. 36, 1.

Kurkela, A., E. S. Fraga, J. Schaffner-Bielich, and A. Vuorinen,2014, Astrophys. J. 789, 127.

Kurkela, A., P. Romatschke, and A. Vuorinen, 2010, Phys. Rev. D 81,105021.

Lacombe, M., et al., 1980, Phys. Rev. C 21, 861.Lacour, A., J. Oller, and U.-G. Meißner, 2011, Ann. Phys.(Amsterdam) 326, 241.

Lagaris, I., and V. Pandharipande, 1981, Nucl. Phys. A 359, 331.Lähde, T. A., et al., 2014, Phys. Lett. B 732, 110.Lalazissis, G., S. Karatzikos, M. Serra, T. Otsuka, and P. Ring, 2009,Phys. Rev. C 80, 041301.

Lalazissis, G. A., J. König, and P. Ring, 1997, Phys. Rev. C 55,540.

Lalazissis, G. A., S. Raman, and P. Ring, 1999, At. Data Nucl. DataTables 71, 1.

Lamb, D., J. Lattimer, C. Pethick, and D. Ravenhall, 1978, Phys. Rev.Lett. 41, 1623.

Lamb, D., J. Lattimer, C. Pethick, and D. Ravenhall, 1983,Nucl. Phys. A 411, 449.

Landau, L. D., and E. Lifshitz, 1980, Statistical Physics: Part 1(Butterworth-Heinemann, Oxford).

Langanke, K., G. Martínez-Pinedo, J. M. Sampaio, D. J. Dean, W. R.Hix, O. E. B. Messer, A. Mezzacappa, M. Liebendörfer, H. T.Janka, and M. Rampp, 2003, Phys. Rev. Lett. 90, 241102.

Lattimer, J., 1981, Annu. Rev. Nucl. Part. Sci. 31, 337.Lattimer, J., C. Pethick, D. Ravenhall, and D. Lamb, 1985,Nucl. Phys. A 432, 646.

Lattimer, J., and M. Prakash, 2001, Astrophys. J. 550, 426.Lattimer, J., and M. Prakash, 2004, Science 304, 536.Lattimer, J. M., 2012, Annu. Rev. Nucl. Part. Sci. 62, 485.Lattimer, J. M., A. Burrows, and A. Yahil, 1985, Astrophys. J. 288,644.

Lattimer, J. M., and Y. Lim, 2013, Astrophys. J. 771, 51.Lattimer, J. M., C. J. Pethick, D. G. Ravenhall, and D. Q. Lamb,1985, Nucl. Phys. A 432, 646.

Lattimer, J. M., and M. Prakash, 2000, Phys. Rep. 333–334, 121.Lattimer, J. M., and M. Prakash, 2007, Phys. Rep. 442, 109.Lattimer, J. M., and M. Prakash, 2011, in From Nuclei to Stars(World Scientific, Singapore), pp. 275–304.

Lattimer, J. M., and D. N. Schramm, 1974, Astrophys. J. 192, L145.Lattimer, J. M., and A.W. Steiner, 2014, Eur. Phys. J. A 50, 40.Lattimer, J. M., and F. D. Swesty, 1991, Nucl. Phys. A 535, 331.Lattimer, J. M., and F. D. Swesty, 1991, “Lattimer-Swesty EOSWeb Site,” http://www.astro.sunysb.edu/dswesty/lseos.html.

Lawrence, S., J. G. Tervala, P. F. Bedaque, and M. C. Miller, 2015,Astrophys. J. 808, 186.

Leahy, D. A., S. M. Morsink, and Y. Chou, 2011, Astrophys. J.742, 17.

Lee, D., 2009, Prog. Part. Nucl. Phys. 63, 117.Le Fèvre, A., Y. Leifels, W. Reisdorf, J. Aichelin, and C. Hartnack,2016, Nucl. Phys. A 945, 112.

Lejeune, A., U. Lombardo, and W. Zuo, 2000, Phys. Lett. B 477, 45.Lesinski, T., M. Bender, K. Bennaceur, T. Duguet, and J. Meyer,2007, Phys. Rev. C 76, 014312.

Li, B.-A., 2002, Nucl. Phys. A 708, 365.Li, B.-A., and L.-W. Chen, 2005, Phys. Rev. C 72, 064611.Li, B.-A., L.-W. Chen, and C. M. Ko, 2008, Phys. Rep. 464, 113.

Li, B.-A., and X. Han, 2013, Phys. Lett. B 727, 276.Li, B.-A., A. Ramos, G. Verde, and Vidaña, I., 2014, Eds., TopicalIssue on Nuclear Symmetry Energy, Vol. A50 of Eur. Phys. J.(Springer, New York).

Liebendörfer, M., 2011, “AGILE-IDSA Web Site,” https://physik.unibas.ch/~liebend/download.

Lin, J., F. Özel, D. Chakrabarty, and D. Psaltis, 2010, Astrophys. J.723, 1053.

Lipparini, E., and S. Stringari, 1989, Phys. Rep. 175, 103.Liu, M., N. Wang, Z.-X. Li, and F.-S. Zhang, 2010, Phys. Rev. C 82,064306.

Liu, M.-Q., J. Zhang, and Z.-Q. Luo, 2007, Chin. Phys. 16, 3146.Liu, X.-J., H. Hu, and P. D. Drummond, 2009, Phys. Rev. Lett. 102,160401.

Lombardo, U., and H. Schulze, 2001, Lect. Notes Phys. 578, 30.Lonardoni, D., S. Gandolfi, and F. Pederiva, 2013, Phys. Rev. C 87,041303.

Lonardoni, D., A. Lovato, S. Gandolfi, and F. Pederiva, 2015, Phys.Rev. Lett. 114, 092301.

Lonardoni, D., F. Pederiva, and S. Gandolfi, 2014, Phys. Rev. C 89,014314.

Long, W., N. Van Giai, and J. Meng, 2006, Phys. Lett. B 640, 150.Long, W. H., P. Ring, N. Van Giai, and J. Meng, 2010, Phys. Rev. C81, 024308.

Long, W. H., H. Sagawa, N. Van Giai, and J. Meng, 2007, Phys. Rev.C 76, 034314.

Long, W.-h., J. Meng, N. Van Giai, and S.-G. Zhou, 2004, Phys. Rev.C 69, 034319.

Lopes, L. L., and D. P. Menezes, 2014, Phys. Rev. C 89, 025805.Lorenz, C., D. Ravenhall, and C. Pethick, 1993, Phys. Rev. Lett. 70,379.

Machleidt, R., 2001, Phys. Rev. C 63, 024001.Machleidt, R., and D. Entem, 2011, Phys. Rep. 503, 1.Machleidt, R., K. Holinde, and C. Elster, 1987, Phys. Rep. 149, 1.Machleidt, R., and G.-Q. Li, 1994, Phys. Rep. 242, 5.Maessen, P. M.M., T. A. Rijken, and J. J. de Swart, 1989, Phys. Rev.C 40, 2226.

Magierski, P., A. Bulgac, and P.-H. Heenen, 2003, Nucl. Phys. A 719,C217.

Magierski, P., and P.-H. Heenen, 2002, Phys. Rev. C 65, 045804.Mallik, S., J. De, S. Samaddar, and S. Sarkar, 2008, Phys. Rev. C 77,032201.

Marek, A., and H.-T. Janka, 2009, Astrophys. J. 694, 664.Marek, A., H.-T. Janka, and E. Müller, 2009, Astron. Astrophys. 496,475.

Margueron, J., S. Goriely, M. Grasso, G. Colo, and H. Sagawa, 2009,J. Phys. G 36, 125103.

Margueron, J., S. Goriely, M. Grasso, G. Colo, and H. Sagawa, 2010,Mod. Phys. Lett. A 25, 1771.

Margueron, J., M. Grasso, S. Goriely, G. Colo, and H. Sagawa, 2012,Prog. Theor. Phys. Suppl. 196, 172.

Margueron, J., J. Navarro, and P. Blottiau, 2004, Phys. Rev. C 70,028801.

Margueron, J., and H. Sagawa, 2009, J. Phys. G 36, 125102.Martin, N., and M. Urban, 2015, Phys. Rev. C 92, 015803.Martínez-Pinedo, G., T. Fischer, A. Lohs, and L. Huther, 2012,Phys. Rev. Lett. 109, 251104.

Martínez-Pinedo, G., T. Fischer, and L. Hüther, 2014, J. Phys. G 41,044008.

Maruyama, T., S. Chiba, H.-J. Schulze, and T. Tatsumi, 2008,Phys. Lett. B 659, 192.

Maruyama, T., T. Tatsumi, D. N. Voskresensky, T. Tanigawa, and S.Chiba, 2005, Phys. Rev. C 72, 015802.





http://dx.doi.org/10.1016/0370-1573(78)90081-9

http://dx.doi.org/10.1016/0370-1573(78)90081-9

http://dx.doi.org/10.1088/0004-637X/789/2/127






http://dx.doi.org/10.1016/0375-9474(81)90240-2





http://dx.doi.org/10.1006/adnd.1998.0795




http://dx.doi.org/10.1016/0375-9474(83)90540-7



http://dx.doi.org/10.1016/0375-9474(85)90006-5

http://dx.doi.org/10.1086/319702



http://dx.doi.org/10.1086/162830

http://dx.doi.org/10.1086/162830

http://dx.doi.org/10.1088/0004-637X/771/1/51

http://dx.doi.org/10.1016/0375-9474(85)90006-5

http://dx.doi.org/10.1016/S0370-1573(00)00019-3


http://dx.doi.org/10.1086/181612


http://dx.doi.org/10.1016/0375-9474(91)90452-C

http://www.astro.sunysb.edu/dswesty/lseos.html





http://dx.doi.org/10.1088/0004-637X/808/2/186

http://dx.doi.org/10.1088/0004-637X/742/1/17

http://dx.doi.org/10.1088/0004-637X/742/1/17



http://dx.doi.org/10.1016/S0370-2693(00)00211-2


http://dx.doi.org/10.1016/S0375-9474(02)01018-7




https://physik.unibas.ch/%7Eliebend/download



http://dx.doi.org/10.1088/0004-637X/723/2/1053

http://dx.doi.org/10.1088/0004-637X/723/2/1053

http://dx.doi.org/10.1016/0370-1573(89)90029-X



http://dx.doi.org/10.1088/1009-1963/16/10/055



http://dx.doi.org/10.1007/3-540-44578-1_2



















http://dx.doi.org/10.1016/S0370-1573(87)80002-9

http://dx.doi.org/10.1016/0370-1573(94)90139-2



http://dx.doi.org/10.1016/S0375-9474(03)00921-7

http://dx.doi.org/10.1016/S0375-9474(03)00921-7




http://dx.doi.org/10.1088/0004-637X/694/1/664

http://dx.doi.org/10.1051/0004-6361/200810883

http://dx.doi.org/10.1051/0004-6361/200810883

http://dx.doi.org/10.1088/0954-3899/36/12/125103

http://dx.doi.org/10.1142/S0217732310000290

http://dx.doi.org/10.1143/PTPS.196.172



http://dx.doi.org/10.1088/0954-3899/36/12/125102



http://dx.doi.org/10.1088/0954-3899/41/4/044008

http://dx.doi.org/10.1088/0954-3899/41/4/044008



Maruyama, T., G. Watanabe, and S. Chiba, 2012, Prog. Theor. Exp.Phys. 2012, 01A201.

Maruyama, T., et al., 1998, Phys. Rev. C 57, 655.Masuda, K., T. Hatsuda, and T. Takatsuka, 2013, Prog. Theor. Exp.Phys. 2013, 073D01.

Masuda, K., T. Hatsuda, and T. Takatsuka, 2016, Prog. Theor. Exp.Phys. 2016, 021D01.

Matsuzaki, M., 2006, Phys. Rev. C 73, 028801.Mazurek, T. J., J. M. Lattimer, and G. E. Brown, 1979, Astrophys. J.229, 713.

Meißner, U. G., J. A. Oller, and A. Wirzba, 2002, Ann. Phys.(Amsterdam) 297, 27.

Menezes, D. P., and C. Providência, 2007, arXiv:astro-ph/0703649.Meng, J., et al., 2006, Prog. Part. Nucl. Phys. 57, 470.Mezzacappa, A., 2005, Annu. Rev. Nucl. Part. Sci. 55, 467.Migdal,A. B., E. E.Saperstein,M. A.Troitsky, andD. N.Voskresensky,1990, Phys. Rep. 192, 179.

Miller, M. C., 2013, arXiv:1312.0029.Mishustin, I., L. Satarov, H. Stoecker, and W. Greiner, 2002, Phys.Rev. C 66, 015202.

Miyatsu, T., S. Yamamuro, and K. Nakazato, 2013, Astrophys. J.777, 4.

Möller, P., W. D. Myers, H. Sagawa, and S. Yoshida, 2012, Phys.Rev. Lett. 108, 052501.

Möller, P., J. R. Nix, W. D. Myers, and W. J. Swiatecki, 1995,At. Data Nucl. Data Tables 59, 185.

Moustakidis, C. C., and C. P. Panos, 2009, Phys. Rev. C 79,045806.

Mukhopadhyay, T., and D. N. Basu, 2007, Acta Phys. Pol. B 38,3225.

Mulero, Á., 2008, Ed., Theory and Simulation of Hard-Sphere Fluidsand Related Systems, Vol. 753 of Lecture Notes in Physics(Springer-Verlag, Berlin).

Müller, D., M. Buballa, and J. Wambach, 2013, Eur. Phys. J. A49, 96.

Müller, H., and B. D. Serot, 1995, Phys. Rev. C 52, 2072.Müller, H., and B. D. Serot, 1996, Nucl. Phys. A 606, 508.Müther, H., and A. Polls, 2000, Prog. Part. Nucl. Phys. 45, 243.Myers, W. D., andW. J. Swiatecki, 1990, Ann. Phys. (N.Y.) 204, 401.Myers, W. D., and W. J. Swiatecki, 1994, Lawrence BerkeleyLaboratory Report No. LBL-36803.

Myers, W. D., and W. J. Swiatecki, 1996, Nucl. Phys. A 601, 141.Nadyozhin, D. K., and A. V. Yudin, 2004, Astron. Lett. 30, 634.Nadyozhin, D. K., and A. V. Yudin, 2005, Astron. Lett. 31, 271.Nagae, T., et al., 1998, Phys. Rev. Lett. 80, 1605.Nagels, M., T. Rijken, and J. de Swart, 1977, Phys. Rev. D 15, 2547.Nagels, M., T. Rijken, and J. de Swart, 1978, Phys. Rev. D 17, 768.Nagels, M., T. A. Rijken, and Y. Yamamoto, 2014, arXiv:1408.4825.Nakada, H., 2003, Phys. Rev. C 68, 014316.Nakazato, K., S. Furusawa, K. Sumiyoshi, A. Ohnishi, S. Yamada,and H. Suzuki, 2012, Astrophys. J. 745, 197.

Nakazato, K., K. Iida, and K. Oyamatsu, 2011, Phys. Rev. C 83,065811.

Nakazato, K., K. Oyamatsu, and S. Yamada, 2009, Phys. Rev. Lett.103, 132501.

Nakazato, K., K. Sumiyoshi, H. Suzuki, and S. Yamada, 2010, Phys.Rev. D 81, 083009.

Nakazato, K., K. Sumiyoshi, and S. Yamada, 2008, Phys. Rev. D 77,103006.

Nakazato, K., K. Sumiyoshi, and S. Yamada, 2010, Astrophys. J.721, 1284.

Nakazato, K., K. Sumiyoshi, and S. Yamada, 2013, Astron.Astrophys. 558, A50.

Nakazawa, K. (KEK-E176 and J-PARC-E07 Collaborations), 2010,Nucl. Phys. A 835, 207.

Nambu, Y., and G. Jona-Lasinio, 1961a, Phys. Rev. 122, 345.Nambu, Y., and G. Jona-Lasinio, 1961b, Phys. Rev. 124, 246.Napolitani, P., P. Chomaz, F. Gulminelli, and K. Hasnaoui, 2007,Phys. Rev. Lett. 98, 131102.

Natowitz, J., et al., 2010, Phys. Rev. Lett. 104, 202501.Navarro, J., and A. Polls, 2013, Phys. Rev. C 87, 044329.Navarro Pérez, R., J. E. Amaro, and E. R. Arriola, 2015, Phys. Rev. C91, 054002.

Navarro Pérez, R., J. E. Amaro, and E. Ruiz Arriola, 2013, Phys. Rev.C 88, 024002 [88, 069902(E) (2013)].

Nazarewicz, W., P. G. Reinhard, W. Satula, and D. Vretenar, 2014,Eur. Phys. J. A 50, 20.

Negele, J. W., and D. Vautherin, 1972, Phys. Rev. C 5, 1472.Negele, J. W., and D. Vautherin, 1973, Nucl. Phys. A 207, 298.Newton, W., and J. Stone, 2009, Phys. Rev. C 79, 055801.Newton, W. G., and B.-a. Li, 2009, Phys. Rev. C 80, 065809.Nickel, D., R. Alkofer, and J. Wambach, 2006, Phys. Rev. D 74,114015.

Nickel, D., J. Wambach, and R. Alkofer, 2006, Phys. Rev. D 73,114028.

Nicotra, O. E., M. Baldo, G. Burgio, and H.-J. Schulze, 2006a,Astron. Astrophys. 451, 213.

Nicotra, O. E., M. Baldo, G. F. Burgio, and H.-J. Schulze, 2006b,Phys. Rev. D 74, 123001.

Nikolaus, B., T. Hoch, and D. Madland, 1992, Phys. Rev. C 46,1757.

Nikšić, T., T. Marketin, D. Vretenar, N. Paar, and P. Ring, 2005,Phys. Rev. C 71, 014308.

Nikšić, T., D. Vretenar, P. Finelli, and P. Ring, 2002, Phys. Rev. C 66,024306.

Nikšić, T., D. Vretenar, and P. Ring, 2011, Prog. Part. Nucl. Phys. 66,519.

Nishimura, N., et al., 2012, Astrophys. J. 758, 9.Nozawa, T., N. Stergioulas, E. Gourgoulhon, and Y. Eriguchi, 1998,Astron. Astrophys. Suppl. Ser. 132, 431.

O’Connor, E., D. Gazit, C. Horowitz, A. Schwenk, and N. Barnea,2007, Phys. Rev. C 75, 055803.

O’Connor, E., and C. D. Ott, 2008, http://www.stellarcollapse.org.O’Connor, E., and C. D. Ott, 2011, Astrophys. J. 730, 70.O’Connor, E., and C. D. Ott, 2013, Astrophys. J. 762, 126.Oechslin, R., H.-T. Janka, and A. Marek, 2007, Astron. Astrophys.467, 395.

Oertel, M., A. Fantina, and J. Novak, 2012, Phys. Rev. C 85,055806.

Oertel, M., F. Gulminelli, C. Providência, and A. R. Raduta, 2016,Eur. Phys. J. A 52, 50.

Oertel, M., C. Providência, F. Gulminelli, and A. R. Raduta, 2015,J. Phys. G 42, 075202.

Oertel, M., and M. Urban, 2008, Phys. Rev. D 77, 074015.Ogata, S., H. Iyetomi, S. Ichimaru, and H. M. Van Horn, 1993,Phys. Rev. E 48, 1344.

Okamoto, M., T. Maruyama, K. Yabana, and T. Tatsumi, 2012,Phys. Lett. B 713, 284.

Oller, J., A. Lacour, and U.-G. Meißner, 2010, J. Phys. G 37, 015106.Ono, A., H. Horiuchi, T. Maruyama, and A. Ohnishi, 1992,Prog. Theor. Phys. 87, 1185.

Onsi, M., et al., 2008, Phys. Rev. C 77, 065805.Oppenheimer, J. R., and G. M. Volkoff, 1939, Phys. Rev. 55, 374.Ott, C. D., 2009, Classical Quantum Gravity 26, 063001.Oyamatsu, K., 1993, Nucl. Phys. A 561, 431.Oyamatsu, K., 1994, Nucl. Phys. A 578, 181.






http://dx.doi.org/10.1093/ptep/ptt045

http://dx.doi.org/10.1093/ptep/ptt045

http://dx.doi.org/10.1093/ptep/ptv187

http://dx.doi.org/10.1093/ptep/ptv187


http://dx.doi.org/10.1086/157006

http://dx.doi.org/10.1086/157006



http://arXiv.org/abs/astro-ph/0703649



http://dx.doi.org/10.1016/0370-1573(90)90132-L




http://dx.doi.org/10.1088/0004-637X/777/1/4

http://dx.doi.org/10.1088/0004-637X/777/1/4









http://dx.doi.org/10.1016/0375-9474(96)00187-X

http://dx.doi.org/10.1016/S0146-6410(00)00105-8

http://dx.doi.org/10.1016/0003-4916(90)90395-5

http://dx.doi.org/10.1016/0375-9474(95)00509-9

http://dx.doi.org/10.1134/1.1795952

http://dx.doi.org/10.1134/1.1896071






http://dx.doi.org/10.1088/0004-637X/745/2/197









http://dx.doi.org/10.1088/0004-637X/721/2/1284

http://dx.doi.org/10.1088/0004-637X/721/2/1284

http://dx.doi.org/10.1051/0004-6361/201322231

http://dx.doi.org/10.1051/0004-6361/201322231














http://dx.doi.org/10.1016/0375-9474(73)90349-7







http://dx.doi.org/10.1051/0004-6361:20053670









http://dx.doi.org/10.1088/0004-637X/758/1/9

http://dx.doi.org/10.1051/aas:1998304


http://www.stellarcollapse.org



http://dx.doi.org/10.1088/0004-637X/730/2/70

http://dx.doi.org/10.1088/0004-637X/762/2/126

http://dx.doi.org/10.1051/0004-6361:20066682

http://dx.doi.org/10.1051/0004-6361:20066682




http://dx.doi.org/10.1088/0954-3899/42/7/075202




http://dx.doi.org/10.1088/0954-3899/37/1/015106

http://dx.doi.org/10.1143/ptp/87.5.1185



http://dx.doi.org/10.1088/0264-9381/26/6/063001

http://dx.doi.org/10.1016/0375-9474(93)90020-X

http://dx.doi.org/10.1016/0375-9474(94)90975-X

Oyamatsu, K., M.-a. Hashimoto, and M. Yamada, 1984, Prog. Theor.Phys. 72, 373.

Oyamatsu, K., and K. Iida, 2007, Phys. Rev. C 75, 015801.Özel, F., G. Baym, and T. Güver, 2010, Phys. Rev. D 82, 101301.Özel, F., A. Gould, and T. Güver, 2012, Astrophys. J. 748, 5.Özel, F., and D. Psaltis, 2009, Phys. Rev. D 80, 103003.Özel, F., D. Psaltis, T. Güver, G. Baym, C. Heinke, and S. Guillot,2016, Astrophys. J. 820, 28.

Paar, N., C. C. Moustakidis, T. Marketin, D. Vretenar, and G.Lalazissis, 2014, Phys. Rev. C 90, 011304.

Page, D., U. Geppert, and F. Weber, 2006, Nucl. Phys. A 777, 497.Page, D., J. M. Lattimer, M. Prakash, and A.W. Steiner, 2014,Novel Superfluids: Volume 2, International Series of Monographson Physics (Oxford University Press, New York), Chap. 21,pp. 505–579.

Page, D., M. Prakash, J. M. Lattimer, and A.W. Steiner, 2011,Phys. Rev. Lett. 106, 081101.

Pais, A., and G. Uhlenbeck, 1959, Phys. Rev. 116, 250.Pais, H., S. Chiacchiera, and C. Providência, 2015, Phys. Rev. C 91,055801.

Pais, H., W. G. Newton, and J. R. Stone, 2014, Phys. Rev. C 90,065802.

Pais, H., and J. R. Stone, 2012, Phys. Rev. Lett. 109, 151101.Pandharipande, V., and R. B. Wiringa, 1979, Rev. Mod. Phys. 51,821.

Pannarale, F., L. Rezzolla, F. Ohme, and J. S. Read, 2011, Phys. Rev.D 84, 104017.

Papakonstantinou, P., J. Margueron, F. Gulminelli, and A. R. Raduta,2013, Phys. Rev. C 88, 045805.

Pastore, A., et al., 2014, Phys. Rev. C 90, 025804.Pavón Valderrama, M., and D. R. Phillips, 2015, Phys. Rev. Lett. 114,082502.

Pearson, J., N. Chamel, S. Goriely, and C. Ducoin, 2012, Phys. Rev.C 85, 065803.

Pearson, J., N. Chamel, A. Pastore, and S. Goriely, 2015, Phys. Rev.C 91, 018801.

Pearson, J., S. Goriely, and N. Chamel, 2011, Phys. Rev. C 83,065810.

Perego, A., M. Hempel, C. Fröhlich, K. Ebinger, M. Eichler, J.Casanova, M. Liebendörfer, and F.-K. Thielemann, 2015,Astrophys. J. 806, 275.

Perego, A., S. Rosswog, R. M. Cabezón, O. Korobkin, R. Käppeli, A.Arcones, and M. Liebendörfer, 2014, Mon. Not. R. Astron. Soc.443, 3134.

Peres, B., M. Oertel, and J. Novak, 2013, Phys. Rev. D 87, 043006.Pethick, C., D. Ravenhall, and C. Lorenz, 1995, Nucl. Phys. A 584,675.

Pethick, C., and V. Thorsson, 1997, Phys. Rev. D 56, 7548.Pethick, C. J., and D. G. Ravenhall, 1988, Ann. Phys. (N.Y.) 183,131.

Piarulli, M., L. Girlanda, R. Schiavilla, R. Navarro Pérez, J. E.Amaro, and E. Ruiz Arriola, 2015, Phys. Rev. C 91, 024003.

Piekarewicz, J., 2004, Phys. Rev. C 69, 041301.Piekarewicz, J., 2010, J. Phys. G 37, 064038.Piekarewicz, J., B. K. Agrawal, G. Colò, W. Nazarewicz, N. Paar,P.-G. Reinhard, X. Roca-Maza, and D. Vretenar, 2012, Phys. Rev.C 85, 041302.

Piekarewicz, J., and G. Toledo Sanchez, 2012, Phys. Rev. C 85,015807.

Pieper, S. C., 2008, Riv. Nuovo Cimento 31, 709.Pieper, S. C., V. Pandharipande, R. B. Wiringa, and J. Carlson, 2001,Phys. Rev. C 64, 014001.

Podsiadlowski, P., et al., 2005, Mon. Not. R. Astron. Soc. 361, 1243.

Polinder, H., J. Haidenbauer, and U.-G. Meißner, 2006, Nucl. Phys.A 779, 244.

Politzer, H. D., 1974, Phys. Rep. 14, 129.Pollock, E., and J. Hansen, 1973, Phys. Rev. A 8, 3110.Pons, J. A., J. A. Miralles, M. Prakash, and J. M. Lattimer, 2001,Astrophys. J. 553, 382.

Pons, J. A., S. Reddy, P. J. Ellis, M. Prakash, and J. M. Lattimer,2000, Phys. Rev. C 62, 035803.

Pons, J. A., S. Reddy, M. Prakash, J. M. Lattimer, and J. A. Miralles,1999, Astrophys. J. 513, 780.

Pons, J. A., A.W. Steiner, M. Prakash, and J. M. Lattimer, 2001,Phys. Rev. Lett. 86, 5223.

Pons, J. A., D. Viganò, and N. Rea, 2013, Nat. Phys. 9, 431.Posselt, B., G. Pavlov, V. Suleimanov, and O. Kargaltsev, 2013,Astrophys. J. 779, 186.

Potekhin, A., 2014, Phys. Usp. 57, 735.Potekhin, A., A. Fantina, N. Chamel, J. Pearson, and S. Goriely,2013, Astron. Astrophys. 560, A48.

Potekhin, A., and P. Haensel, 2013, “Equations of State,” http://www.ioffe.ru/astro/NSG/nseoslist.html.

Potekhin, A. Y., 2010, Phys. Usp. 53, 1235, http://stacks.iop.org/1063‑7869/53/i=12/a=R03.

Potekhin, A. Y., and G. Chabrier, 2000, Phys. Rev. E 62, 8554.Potekhin, A. Y., and G. Chabrier, 2010, Contrib. Plasma Phys.50, 82.

Potekhin, A. Y., G. Chabrier, A. I. Chugunov, H. E. DeWitt, and F. J.Rogers, 2009, Phys. Rev. E 80, 047401.

Potekhin, A. Y., G. Chabrier, and F. J. Rogers, 2009, Phys. Rev. E 79,016411.

Potekhin, A. Y., J. A. Pons, and D. Page, 2015, Space Sci. Rev. 191,239.

Poutanen, J., et al., 2014, Mon. Not. R. Astron. Soc. 442, 3777.Prakash, M., J. R. Cooke, and J. M. Lattimer, 1995, Phys. Rev. D 52,661.

Prakash, M., P. J. Ellis, and J. I. Kapusta, 1992, Phys. Rev. C 45,2518.

Prakash, M., et al., 1997, Phys. Rep. 280, 1.Pratt, S., P. Siemens, and Q. Usmani, 1987, Phys. Lett. B 189, 1.Providência, C., L. Brito, S. Avancini, D. Menezes, and P. Chomaz,2006, Phys. Rev. C 73, 025805.

Providência, C., L. Brito, A. Santos, D. Menezes, and S. Avancini,2006, Phys. Rev. C 74, 045802.

Pudliner, B., V. Pandharipande, J. Carlson, S. C. Pieper, and R. B.Wiringa, 1997, Phys. Rev. C 56, 1720.

Pudliner, B., V. Pandharipande, J. Carlson, and R. B. Wiringa, 1995,Phys. Rev. Lett. 74, 4396.

Qin, L., et al., 2012, Phys. Rev. Lett. 108, 172701.Qin, S.-x., L. Chang, H. Chen, Y.-x. Liu, and C. D. Roberts, 2011,Phys. Rev. Lett. 106, 172301.

Quentin, P., and H. Flocard, 1978, Annu. Rev. Nucl. Part. Sci. 28,523.

Raduta, A., F. Aymard, and F. Gulminelli, 2014, Eur. Phys. J. A 50, 24.Raduta, A., and F. Gulminelli, 2009, Phys. Rev. C 80, 024606.Raduta, A., and F. Gulminelli, 2010, Phys. Rev. C 82, 065801.Raduta, A. R., F. Gulminelli, and M. Oertel, 2016, Phys. Rev. C 93,025803.

Ratti, C., M. A. Thaler, and W. Weise, 2006, Phys. Rev. D 73,014019.

Rauscher, T., 2003, Astrophys. J. Suppl. Ser. 147, 403.Rauscher, T., and F.-K. Thielemann, 2000, At. Data Nucl. DataTables 75, 1.

Rauscher, T., and F.-K. Thielemann, 2001, At. Data Nucl. DataTables 79, 47.







http://dx.doi.org/10.1088/0004-637X/748/1/5


http://dx.doi.org/10.3847/0004-637X/820/1/28
























http://dx.doi.org/10.1088/0004-637X/806/2/275




http://dx.doi.org/10.1016/0375-9474(94)00506-I

http://dx.doi.org/10.1016/0375-9474(94)00506-I


http://dx.doi.org/10.1016/0003-4916(88)90249-7

http://dx.doi.org/10.1016/0003-4916(88)90249-7



http://dx.doi.org/10.1088/0954-3899/37/6/064038





http://dx.doi.org/10.1393/ncr/i2009-10039-1


http://dx.doi.org/10.1111/j.1365-2966.2005.09253.x



http://dx.doi.org/10.1016/0370-1573(74)90014-3


http://dx.doi.org/10.1086/320642


http://dx.doi.org/10.1086/306889


http://dx.doi.org/10.1038/nphys2640

http://dx.doi.org/10.1088/0004-637X/779/2/186

http://dx.doi.org/10.3367/UFNe.0184.201408a.0793

http://dx.doi.org/10.1051/0004-6361/201321697

http://www.ioffe.ru/astro/NSG/nseoslist.html




http://stacks.iop.org/1063-7869/53/i=12/a=R03



http://dx.doi.org/10.3367/UFNe.0180.201012c.1279







http://dx.doi.org/10.1007/s11214-015-0180-9

http://dx.doi.org/10.1007/s11214-015-0180-9






http://dx.doi.org/10.1016/S0370-1573(96)00023-3

http://dx.doi.org/10.1016/0370-2693(87)91259-7
















http://dx.doi.org/10.1086/375733





Ravenhall, D., C. Bennett, and C. Pethick, 1972, Phys. Rev. Lett. 28,978.

Ravenhall, D., C. Pethick, and J. Wilson, 1983, Phys. Rev. Lett. 50,2066.

Ray, L., 1979, Phys. Rev. C 19, 1855.Ray, L., and P. Hodgson, 1979, Phys. Rev. C 20, 2403.Read, J. S., B. D. Lackey, B. J. Owen, and J. L. Friedman, 2009,Phys. Rev. D 79, 124032.

Read, J. S., et al., 2013, Phys. Rev. D 88, 044042.Reinhard, P., 1989, Rep. Prog. Phys. 52, 439.Reinhard, P., M. Rufa, J. Maruhn, W. Greiner, and J. Friedrich, 1986,Z. Phys. A 323, 13.

Reinhard, P.-G., and W. Nazarewicz, 2010, Phys. Rev. C 81, 051303.Reinhard, P.-G., and W. Nazarewicz, 2013, Phys. Rev. C 87, 014324.Reisdorf, W., et al. (FOPI Collaboration), 2007, Nucl. Phys. A 781,459.

Reisdorf, W., et al. (FOPI Collaboration), 2012, Nucl. Phys. A876, 1.

Rijken, T., 2006, Phys. Rev. C 73, 044007.Rijken, T., and Y. Yamamoto, 2006, Phys. Rev. C 73, 044008.Rijken, T. A., V. G. J. Stoks, and Y. Yamamoto, 1999, Phys. Rev. C59, 21.

Rikovska Stone, J., J. C. Miller, R. Koncewicz, P. D. Stevenson, andM. R. Strayer, 2003, Phys. Rev. C 68, 034324.

Rikovska Stone, J., P. D. Stevenson, J. C. Miller, and M. R. Strayer,2002, Phys. Rev. C 65, 064312.

Rikovska-Stone, J., P. A. Guichon, H. H. Matevosyan, and A.W.Thomas, 2007, Nucl. Phys. A 792, 341.

Ring, P., 1996, Prog. Part. Nucl. Phys. 37, 193.Ring, P., and P. Schuck, 1980, The nuclear many-body problem(Springer-Verlag, Berlin/Heidelberg).

Rios, A., A. Polls, A. Ramos, and H. Müther, 2008, Phys. Rev. C 78,044314.

Rios, A., A. Polls, and I. Vidaña, 2009, Phys. Rev. C 79, 025802.Rischke, D. H., M. I. Gorenstein, H. Stoecker, and W. Greiner, 1991,Z. Phys. C 51, 485.

Roberts, C. D., 2015, “Strong QCD and Dyson-Schwinger equa-tions” in EMS IRMA Lectures in Mathematics and TheoreticalPhysics, edited by K. Ebrahimi-Fard and F. Fauvet (EuropeanMathematical Society Publishing House, Zürich, Switzerland),Vol. 21, p. 355.

Roberts, C. D., R. Cahill, and J. Praschifka, 1988, Ann. Phys. (N.Y.)188, 20.

Roberts, C. D., and S.M. Schmidt, 2000, Prog. Part. Nucl. Phys.45, S1.

Roberts, L., S. Reddy, and G. Shen, 2012, Phys. Rev. C 86, 065803.Roberts, L., et al., 2012, Phys. Rev. Lett. 108, 061103.Roberts, L. F., 2012, Astrophys. J. 755, 126.Roberts, L. F., S. E. Woosley, and R. D. Hoffman, 2010, Astrophys. J.722, 954.

Roca-Maza, X., M. Brenna, B. K. Agrawal, P. F. Bortignon, G. Colò,L.-G. Cao, N. Paar, andD. Vretenar, 2013, Phys. Rev. C 87, 034301.

Roca-Maza, X., M. Centelles, X. Viñas, M. Brenna, G. Colò,B. K. Agrawal, N. Paar, J. Piekarewicz, and D. Vretenar, 2013,Phys. Rev. C 88, 024316.

Roca-Maza, X., M. Centelles, X. Viñas, and M. Warda, 2011,Phys. Rev. Lett. 106, 252501.

Roca-Maza, X., and J. Piekarewicz, 2008, Phys. Rev. C 78, 025807.Roca-Maza, X., J. Piekarewicz, T. Garcia-Galvez, and M. Centelles,2012, in Neutron star crust, edited by J. P. C. Bertulani(Nova Science Publishers, New York).

Roca-Maza, X., X. Viñas, M. Centelles, B. K. Agrawal, G. Colo’, N.Paar, J. Piekarewicz, andD.Vretenar, 2015, Phys.Rev.C 92, 064304.

Roca-Maza, X., X. Viñas, M. Centelles, P. Ring, and P. Schuck, 2011,Phys. Rev. C 84, 054309.

Roggero, A., A. Mukherjee, and F. Pederiva, 2014, Phys. Rev. Lett.112, 221103.

Romani, R.W., A. V. Filippenko, J. M. Silverman, S. B. Cenko, J.Greiner, A. Rau, J. Elliott, and H. J. Pletsch, 2012, Astrophys. J.Lett. 760, L36.

Röpke, G., 2009, Phys. Rev. C 79, 014002.Röpke, G., 2011, Nucl. Phys. A 867, 66.Röpke, G., L. Münchow, and H. Schulz, 1982, Nucl. Phys. A 379,536.

Röpke, G., M. Schmidt, L. Münchow, and H. Schulz, 1983, Nucl.Phys. A 399, 587.

Rosswog, S., 2015, Int. J. Mod. Phys. D 24, 1530012.Rosswog, S., T. Piran, and E. Nakar, 2013, Mon. Not. R. Astron. Soc.430, 2585.

Rotondo, M., J. A. Rueda, R. Ruffini, and S.-S. Xue, 2011,Phys. Rev. C 83, 045805.

Ruester, S. B., M. Hempel, and J. Schaffner-Bielich, 2006, Phys. Rev.C 73, 035804.

Rufa, M., P. Reinhard, J. Maruhn, W. Greiner, and M. Strayer, 1988,Phys. Rev. C 38, 390.

Rusnak, J. J., and R. Furnstahl, 1997, Nucl. Phys. A 627, 495.Russotto, P., et al., 2011, Phys. Lett. B 697, 471.Russotto, P., et al., 2014, Eur. Phys. J. A 50, 38.Russotto, P., et al. (ASY-EOS Collaboration), 2013, J. Phys. Conf.Ser. 420, 012092.

Sagert, I., T. Fischer, M. Hempel, G. Pagliara, J. Schaffner-Bielich,A. Mezzacappa, F.-K. Thielemann, and M. Liebendörfer, 2009,Phys. Rev. Lett. 102, 081101.

Sagert, I., T. Fischer, M. Hempel, G. Pagliara, J. Schaffner-Bielich,F.-K. Thielemann, and M. Liebendörfer, 2010, J. Phys. G 37,094064.

Sagert, I., L. Tolos, D. Chatterjee, J. Schaffner-Bielich, and C. Sturm,2012, Phys. Rev. C 86, 045802.

Sagert, I., et al., 2012, Acta Phys. Pol. B 43, 741.Sagun, V. V., A. I. Ivanytskyi, K. A. Bugaev, and I. N. Mishustin,2014, Nucl. Phys. A 924, 24.

Saha, P. K., et al., 2004, Phys. Rev. C 70, 044613.Salpeter, E., 1961, Astrophys. J. 134, 669.Sammarruca, F., 2010, Int. J. Mod. Phys. E 19, 1259.Sammarruca, F., B. Chen, L. Coraggio, N. Itaco, and R. Machleidt,2012, Phys. Rev. C 86, 054317.

Schaefer, B.-J., M. Wagner, J. Wambach, T. Kuo, and G. Brown,2006, Phys. Rev. C 73, 011001.

Schaffner, J., and I. N. Mishustin, 1996, Phys. Rev. C 53, 1416.Schaffner, J., et al., 1994, Ann. Phys. (N.Y.) 235, 35.Schaffner-Bielich, J., and A. Gal, 2000, Phys. Rev. C 62, 034311.Schaffner-Bielich, J., M. Hanauske, H. Stoecker, and W. Greiner,2002, Phys. Rev. Lett. 89, 171101.

Scheidegger, S., R. Käppeli, S. C. Whitehouse, T. Fischer, and M.Liebendörfer, 2010, Astron. Astrophys. 514, A51.

Schmidt, K. E., and S. Fantoni, 1999, Phys. Lett. B 446, 99.Schmidt, M., G. Röpke, and H. Schulz, 1990, Ann. Phys. (N.Y.)202, 57.

Schmidt, S., D. Blaschke, and Y. Kalinovsky, 1994, Phys. Rev. C 50,435.

Schneider, A., D. Berry, C. Briggs, M. Caplan, and C. Horowitz,2014, Phys. Rev. C 90, 055805.

Schneider, A., C. Horowitz, J. Hughto, and D. Berry, 2013,Phys. Rev. C 88, 065807.

Schuetrumpf, B., et al., 2013a, J. Phys. Conf. Ser. 426, 012009.Schuetrumpf, B., et al., 2013b, Phys. Rev. C 87, 055805.











http://dx.doi.org/10.1088/0034-4885/52/4/002

http://dx.doi.org/10.1007/BF01294551














http://dx.doi.org/10.1016/0146-6410(96)00054-3




http://dx.doi.org/10.1007/BF01548574

http://dx.doi.org/10.1016/0003-4916(88)90090-5

http://dx.doi.org/10.1016/0003-4916(88)90090-5

http://dx.doi.org/10.1016/S0146-6410(00)90011-5

http://dx.doi.org/10.1016/S0146-6410(00)90011-5



http://dx.doi.org/10.1088/0004-637X/755/2/126

http://dx.doi.org/10.1088/0004-637X/722/1/954

http://dx.doi.org/10.1088/0004-637X/722/1/954









http://dx.doi.org/10.1088/2041-8205/760/2/L36

http://dx.doi.org/10.1088/2041-8205/760/2/L36



http://dx.doi.org/10.1016/0375-9474(82)90013-6

http://dx.doi.org/10.1016/0375-9474(82)90013-6

http://dx.doi.org/10.1016/0375-9474(83)90265-8

http://dx.doi.org/10.1016/0375-9474(83)90265-8

http://dx.doi.org/10.1142/S0218271815300128

http://dx.doi.org/10.1093/mnras/sts708

http://dx.doi.org/10.1093/mnras/sts708





http://dx.doi.org/10.1016/S0375-9474(97)00598-8



http://dx.doi.org/10.1088/1742-6596/420/1/012092

http://dx.doi.org/10.1088/1742-6596/420/1/012092


http://dx.doi.org/10.1088/0954-3899/37/9/094064

http://dx.doi.org/10.1088/0954-3899/37/9/094064


http://dx.doi.org/10.5506/APhysPolB.43.741



http://dx.doi.org/10.1086/147194

http://dx.doi.org/10.1142/S0218301310015874







http://dx.doi.org/10.1051/0004-6361/200913220

http://dx.doi.org/10.1016/S0370-2693(98)01522-6

http://dx.doi.org/10.1016/0003-4916(90)90340-T

http://dx.doi.org/10.1016/0003-4916(90)90340-T





http://dx.doi.org/10.1088/1742-6596/426/1/012009


Schuetrumpf, B., et al., 2015, Phys. Rev. C 91, 025801.Schulze, H.-J., and T. Rijken, 2011, Phys. Rev. C 84, 035801.Schwenk, A., 2005, J. Phys. G 31, S1273.Sebille, F., V. de la Mota, and S. Figerou, 2011, Phys. Rev. C 84,055801.

Sebille, F., S. Figerou, and V. de la Mota, 2009, Nucl. Phys. A822, 51.

Sechi-Zorn, B., B. Kehoe, J. Twitty, and R. Burnstein, 1968,Phys. Rev. 175, 1735.

Sedrakian, A., 2007, Prog. Part. Nucl. Phys. 58, 168.Sedrakian, A., 2013, Astron. Astrophys. 555, L10.Seitenzahl, I. R., D. M. Townsley, F. Peng, and J. W. Truran, 2009,At. Data Nucl. Data Tables 95, 96.

Sekiguchi, Y., K. Kiuchi, K. Kyutoku, and M. Shibata, 2011,Phys. Rev. Lett. 107, 211101.

Sekiguchi, Y., K. Kiuchi, K. Kyutoku, and M. Shibata, 2015,Phys. Rev. D 91, 064059.

Sekiguchi, Y., and M. Shibata, 2011, Astrophys. J. 737, 6.Sellahewa, R., and A. Rios, 2014, Phys. Rev. C 90, 054327.Serot, B. D., 1992, Rep. Prog. Phys. 55, 1855.Serot, B. D., and J. D. Walecka, 1986, Adv. Nucl. Phys. 16, 1.Serot, B. D., and J. D. Walecka, 1997, Int. J. Mod. Phys. E 06, 515.Sharma, B. K., M. Centelles, X. Viñas, M. Baldo, and G. F. Burgio,2015, Astron. Astrophys. 584, A103.

Sharma, M. M., 2009, Nucl. Phys. A 816, 65.Shen, G., and C. Horowitz, 2010, “Gang Shen EOSWeb Site,” http://cecelia.physics.indiana.edu/gang_shen_eos.

Shen, G., C. Horowitz, and S. Teige, 2010, Phys. Rev. C 82, 015806.Shen, G., C. J. Horowitz, and E. O’Connor, 2011, Phys. Rev. C 83,065808.

Shen, G., C. J. Horowitz, and S. Teige, 2011, Phys. Rev. C 83,035802.

Shen, H., H. Toki, K. Oyamatsu, and K. Sumiyoshi, 1998a,Nucl. Phys. A 637, 435.

Shen, H., H. Toki, K. Oyamatsu, and K. Sumiyoshi, 1998b,Prog. Theor. Phys. 100, 1013.

Shen, H., H. Toki, K. Oyamatsu, and K. Sumiyoshi, 2011, Astrophys.J. Suppl. Ser. 197, 20.

Shetty, D. V., S. J. Yennello, and G. A. Souliotis, 2007, Phys. Rev. C76, 024606 [76, 039902(E) (2007)].

Shibata, M., and K. Taniguchi, 2011, Living Rev. Relativ. 14, 6.Shlomo, S., V. M. Kolomietz, and G. Colò, 2006, Eur. Phys. J. A30, 23.

Shternin, P. S., D. G. Yakovlev, C. O. Heinke, W. C. Ho, and D. J.Patnaude, 2011, Mon. Not. R. Astron. Soc. 412, L108.

Sil, T., et al., 2002, Phys. Rev. C 66, 045803.Skyrme, T., 1956, Philos. Mag. 1, 1043.Skyrme, T., 1959, Nucl. Phys. 9, 615.Sonoda, H., G. Watanabe, K. Sato, K. Yasuoka, and T. Ebisuzaki,2008, Phys. Rev. C 77, 035806.

Sotani, H., K. Iida, K. Oyamatsu, and A. Ohnishi, 2014, Prog. Theor.Exp. Phys. 2014, 051E01.

Sotani, H., K. Nakazato, K. Iida, and K. Oyamatsu, 2012, Phys. Rev.Lett. 108, 201101.

Sotani, H., K. Nakazato, K. Iida, and K. Oyamatsu, 2013a, Mon. Not.R. Astron. Soc. Lett. 428, L21.

Sotani, H., K. Nakazato, K. Iida, and K. Oyamatsu, 2013b, Mon. Not.R. Astron. Soc. 434, 2060.

Souza, S. R., A.W. Steiner, W. G. Lynch, R. Donangelo, and M. A.Famiano, 2009, Astrophys. J. 707, 1495.

Stein, H., K. Morawetz, and G. Röpke, 1997, Phys. Rev. A 55, 1945.Stein, H., A. Schnell, T. Alm, and G. Röpke, 1995, Z. Phys. A 351,295.

Steiner, A., M. Prakash, and J. M. Lattimer, 2000, Phys. Lett. B 486,239.

Steiner, A.W., 2008, Phys. Rev. C 77, 035805.Steiner, A.W., and S. Gandolfi, 2012, Phys. Rev. Lett. 108,081102.

Steiner, A.W., M. Hempel, and T. Fischer, 2013, Astrophys. J.774, 17.

Steiner, A.W., J. M. Lattimer, and E. F. Brown, 2010, Astrophys. J.722, 33.

Steiner, A.W., J. M. Lattimer, and E. F. Brown, 2013, Astrophys. J.765, L5.

Steiner, A.W., M. Prakash, J. M. Lattimer, and P. J. Ellis, 2005,Phys. Rep. 411, 325.

Steiner, A.W., and A. L. Watts, 2009, Phys. Rev. Lett. 103, 181101.Stergioulas, N., 1996, “RNS: Rapidly Rotating Neutron Stars,” http://www.gravity.phys.uwm.edu/rns.

Stoitsov, M., et al., 2010, Phys. Rev. C 82, 054307.Stoks, V., R. Kompl, M. Rentmeester, and J. de Swart, 1993,Phys. Rev. C 48, 792.

Stone, J., 2005, J. Phys. G 31, R211.Stone, J., and P.-G. Reinhard, 2007, Prog. Part. Nucl. Phys. 58, 587.Stone, J., N. Stone, and S. Moszkowski, 2014, Phys. Rev. C 89,044316.

Strachan, A., and C. Dorso, 1997, Phys. Rev. C 56, 995.Sturm, C. T., et al. (KAOS Collaboration), 2001, Phys. Rev. Lett.86, 39.

Sugahara, Y., and H. Toki, 1994, Nucl. Phys. A 579, 557.Sugimura, H., et al. (J-PARC E10 Collaboration), 2014, Phys. Lett. B729, 39.

Sullivan, C., E. O’Connor, R. G. T. Zegers, T. Grubb, and S. M.Austin, 2016, Astrophys. J. 816, 44.

Sumiyoshi, K., 1998, Home page of Relativistic EoS, http://user.numazu‑ct.ac.jp/~sumi/eos/index.html.

Sumiyoshi, K., C. Ishizuka, A. Ohnishi, S. Yamada, and H. Suzuki,2009, Astrophys. J. Lett. 690, L43.

Sumiyoshi, K., K. Oyamatsu, and H. Toki, 1995, Nucl. Phys. A 595,327.

Sumiyoshi, K., and G. Röpke, 2008, Phys. Rev. C 77, 055804.Sumiyoshi, K., S. Yamada, and H. Suzuki, 2007, Astrophys. J. 667,382.

Sumiyoshi, K., S. Yamada, H. Suzuki, H. Shen, S. Chiba, and H.Toki, 2005, Astrophys. J. 629, 922.

Sun, Z. Y., et al., 2010, Phys. Rev. C 82, 051603.Suwa, Y., 2014, Publ. Astron. Soc. Jpn. 66, L1.Suwa, Y., et al., 2013, Astrophys. J. 764, 99.Takami, K., L. Rezzolla, and L. Baiotti, 2014, Phys. Rev. Lett. 113,091104.

Tamii, A., P. von Neumann-Cosel, and I. Poltoratska, 2014,Eur. Phys. J. A 50, 28.

Tamii, A., et al., 2011, Phys. Rev. Lett. 107, 062502.Tanvir, N. R., A. J. Levan, A. S. Fruchter, J. Hjorth, R. A. Hounsell,K. Wiersema, and R. L. Tunnicliffe, 2013, Nature (London) 500,547.

Tarbert, C., et al., 2014, Phys. Rev. Lett. 112, 242502.Tauris, T. M., N. Langer, T. J. Moriya, P. Podsiadlowski, S.-C. Yoon,and S. I. Blinnikov, 2013, Astrophys. J. Lett. 778, L23.

Terashima, S., et al., 2008, Phys. Rev. C 77, 024317.Ter Haar, B., and R. Malfliet, 1987, Phys. Rep. 149, 207.Tews, I., T. Krüger, K. Hebeler, and A. Schwenk, 2013, Phys. Rev.Lett. 110, 032504.

Thomas, A., D. Whittenbury, J. Carroll, K. Tsushima, and J. Stone,2013, EPJ Web Conf. 63, 03004.

Thouless, D. J., 1960, Ann. Phys. (N.Y.) 10, 553.





http://dx.doi.org/10.1088/0954-3899/31/8/005







http://dx.doi.org/10.1051/0004-6361/201321541




http://dx.doi.org/10.1088/0004-637X/737/1/6


http://dx.doi.org/10.1088/0034-4885/55/11/001

http://dx.doi.org/10.1142/S0218301397000299

http://dx.doi.org/10.1051/0004-6361/201526642


http://cecelia.physics.indiana.edu/gang_shen_eos










http://dx.doi.org/10.1016/S0375-9474(98)00236-X


http://dx.doi.org/10.1088/0067-0049/197/2/20

http://dx.doi.org/10.1088/0067-0049/197/2/20







http://dx.doi.org/10.1111/j.1745-3933.2011.01015.x


http://dx.doi.org/10.1080/14786435608238186

http://dx.doi.org/10.1016/0029-5582(58)90345-6


http://dx.doi.org/10.1093/ptep/ptu052




http://dx.doi.org/10.1093/mnrasl/sls006

http://dx.doi.org/10.1093/mnrasl/sls006

http://dx.doi.org/10.1093/mnras/stt1152

http://dx.doi.org/10.1093/mnras/stt1152

http://dx.doi.org/10.1088/0004-637X/707/2/1495


http://dx.doi.org/10.1007/BF01290911

http://dx.doi.org/10.1007/BF01290911

http://dx.doi.org/10.1016/S0370-2693(00)00780-2

http://dx.doi.org/10.1016/S0370-2693(00)00780-2




http://dx.doi.org/10.1088/0004-637X/774/1/17

http://dx.doi.org/10.1088/0004-637X/774/1/17

http://dx.doi.org/10.1088/0004-637X/722/1/33

http://dx.doi.org/10.1088/0004-637X/722/1/33

http://dx.doi.org/10.1088/2041-8205/765/1/L5

http://dx.doi.org/10.1088/2041-8205/765/1/L5



http://www.gravity.phys.uwm.edu/rns








http://dx.doi.org/10.1088/0954-3899/31/11/R01







http://dx.doi.org/10.1016/0375-9474(94)90923-7



http://dx.doi.org/10.3847/0004-637X/816/1/44

http://user.numazu-ct.ac.jp/%7Esumi/eos/index.html





http://dx.doi.org/10.1088/0004-637X/690/1/L43

http://dx.doi.org/10.1016/0375-9474(95)00388-5

http://dx.doi.org/10.1016/0375-9474(95)00388-5


http://dx.doi.org/10.1086/520876

http://dx.doi.org/10.1086/520876

http://dx.doi.org/10.1086/431788


http://dx.doi.org/10.1093/pasj/pst030

http://dx.doi.org/10.1088/0004-637X/764/1/99








http://dx.doi.org/10.1088/2041-8205/778/2/L23


http://dx.doi.org/10.1016/0370-1573(87)90085-8



http://dx.doi.org/10.1051/epjconf/20136303004

http://dx.doi.org/10.1016/0003-4916(60)90122-6

Todd-Rutel, B., and J. Piekarewicz, 2005, Phys. Rev. Lett. 95,122501.

Togashi, H., and M. Takano, 2013, Nucl. Phys. A 902, 53.Togashi, H., M. Takano, K. Sumiyoshi, and K. Nakazato, 2014,Prog. Theor. Exp. Phys. 2014, 023D05.

Toki, H., D. Hirata, Y. Sugahara, K. Sumiyoshi, and I. Tanihata,1995, Nucl. Phys. A 588, c357.

Tolman, R. C., 1939, Phys. Rev. 55, 364.Tolos, L., B. Friman, and A. Schwenk, 2008, Nucl. Phys. A 806, 105.Trippa, L., G. Colò, and E. Vigezzi, 2008, Phys. Rev. C 77, 061304.Trzcinska, A., et al., 2001, Phys. Rev. Lett. 87, 082501.Tsang, M., et al., 2009, Phys. Rev. Lett. 102, 122701.Tsang, M., et al., 2012, Phys. Rev. C 86, 015803.Tsang, M. B., et al., 2004, Phys. Rev. Lett. 92, 062701.Typel, S., 2005, Phys. Rev. C 71, 064301.Typel, S., 2013, AIP Conf. Proc. 1520, 68.Typel, S., 2014, Phys. Rev. C 89, 064321.Typel, S., 2016, Eur. Phys. J. A 52, 16.Typel, S., and B. A. Brown, 2001, Phys. Rev. C 64, 027302.Typel, S., M. Oertel, and T. Klähn, 2015, Phys. Part. Nucl. 46, 633.Typel, S., G. Röpke, T. Klähn, D. Blaschke, and H. Wolter, 2010,Phys. Rev. C 81, 015803.

Typel, S., T. von Chossy, and H. Wolter, 2003, Phys. Rev. C 67,034002.

Typel, S., and H. Wolter, 1999, Nucl. Phys. A 656, 331.Typel, S., H. Wolter, G. Röpke, and D. Blaschke, 2014, Eur. Phys. J.A 50, 17.

van Dalen, E., G. Colucci, and A. Sedrakian, 2014, Phys. Lett. B 734,383.

van Dalen, E., C. Fuchs, and A. Faessler, 2004, Nucl. Phys. A 744,227.

van Kerkwijk, M., R. Breton, and S. Kulkarni, 2011, Astrophys. J.728, 95.

Vantournhout, K., and H. Feldmeier, 2012, J. Phys. Conf. Ser. 342,012011.

Vantournhout, K., N. Jachowicz, and J. Ryckebusch, 2011,Phys. Rev. C 84, 032801.

Vantournhout, K., T. Neff, H. Feldmeier, N. Jachowicz, and J.Ryckebusch, 2011, Prog. Part. Nucl. Phys. 66, 271.

Vautherin, D., and D. Brink, 1970, Phys. Lett. B 32, 149.Vautherin, D., and D. Brink, 1972, Phys. Rev. C 5, 626.Venugopalan, R., and M. Prakash, 1992, Nucl. Phys. A 546, 718.Verbiest, J., et al., 2008, Astrophys. J. 679, 675.Vidaña, I., 2012, Phys. Rev. C 85, 045808 [90, 029901(E) (2014)].Vidaña, I., D. Logoteta, C. Providençia, A. Polls, and I. Bombaci,2010, arXiv:1004.3958.

Vidaña, I., D. Logoteta, C. Providençia, A. Polls, and I. Bombaci,2011, Europhys. Lett. 94, 11002.

Vidaña, I., A. Polls, A. Ramos, and H. J. Schulze, 2001, Phys. Rev. C64, 044301.

Viñas, X., M. Centelles, X. Roca-Maza, and M. Warda, 2014a, AIPConf. Proc. 1606, 256.

Viñas, X., M. Centelles, X. Roca-Maza, and M. Warda, 2014b,Eur. Phys. J. A 50, 27.

Voskresenskaya, M., and S. Typel, 2012, Nucl. Phys. A 887, 42.Voskresensky, D. N., M. Yasuhira, and T. Tatsumi, 2003, Nucl. Phys.A 723, 291.

Vretenar, D., A. Afanasjev, G. Lalazissis, and P. Ring, 2005,Phys. Rep. 409, 101.

Wada, R., et al., 2012, Phys. Rev. C 85, 064618.Wagner, M., B.-J. Schaefer, J. Wambach, T. Kuo, and G. Brown,2006, Phys. Rev. C 74, 054003.

Walecka, J., 1974, Ann. Phys. (N.Y.) 83, 491.

Walhout, T., R. Cenni, A. Fabrocini, and S. Fantoni, 1996, Phys. Rev.C 54, 1622.

Wanajo, S., et al., 2014, Astrophys. J. 789, L39.Wang, J.-c.,Q.Wang, andD. H.Rischke, 2011, Phys. Lett. B 704, 347.Wang, M., G. Audi, A. H. Wapstra, F. G. Kondev, M. MacCormick,X. Xu, and B. Pfeiffer, 2012, Chin. Phys. C 36, 1603.

Wang, N., and T. Li, 2013, Phys. Rev. C 88, 011301.Wang, N., L. Ou, and M. Liu, 2013, Phys. Rev. C 87, 034327.Warda, M., X. Viñas, X. Roca-Maza, and M. Centelles, 2009,Phys. Rev. C 80, 024316.

Warda, M., X. Viñas, X. Roca-Maza, and M. Centelles, 2010,Phys. Rev. C 81, 054309.

Washiyama, K., et al., 2012, Phys. Rev. C 86, 054309.Watanabe, G., 2007, AIP Conf. Proc. 891, 373.Watanabe, G., and K. Iida, 2003, Phys. Rev. C 68, 045801.Watanabe, G., K. Iida, and K. Sato, 2000, Nucl. Phys. A 676,455.

Watanabe, G., T. Maruyama, K. Sato, K. Yasuoka, and T. Ebisuzaki,2005, Phys. Rev. Lett. 94, 031101.

Watanabe, G., K. Sato, K. Yasuoka, and T. Ebisuzaki, 2002a,Prog. Theor. Phys. Suppl. 146, 638.

Watanabe, G., K. Sato, K. Yasuoka, and T. Ebisuzaki, 2002b,Phys. Rev. C 66, 012801.

Watanabe, G., K. Sato, K. Yasuoka, and T. Ebisuzaki, 2003a,Nucl. Phys. A 718, 700.

Watanabe, G., K. Sato, K. Yasuoka, and T. Ebisuzaki, 2003b,Phys. Rev. C 68, 035806.

Watanabe, G., K. Sato, K. Yasuoka, and T. Ebisuzaki, 2004,Phys. Rev. C 69, 055805.

Watanabe, G., et al., 2009, Phys. Rev. Lett. 103, 121101.Weber, F., 1999, Pulsars as astrophysical laboratories fornuclear and particle physics (Institute of Physics Publishing,Bristol, UK).

Weber, F., 2005, Prog. Part. Nucl. Phys. 54, 193.Weinberg, S., 1990, Phys. Lett. B 251, 288.Weinberg, S., 1991, Nucl. Phys. B 363, 3.Weissenborn, S., D. Chatterjee, and J. Schaffner-Bielich, 2012a,Nucl. Phys. A 881, 62.

Weissenborn, S., D. Chatterjee, and J. Schaffner-Bielich, 2012b,Phys. Rev. C 85, 065802.

Weissenborn, S., I. Sagert, G. Pagliara, M. Hempel, and J. Schaffner-Bielich, 2011, Astrophys. J. 740, L14.

Welke, G. M., M. Prakash, T. T. S. Kuo, S. Das Gupta, and C. Gale,1988, Phys. Rev. C 38, 2101.

Wellenhofer, C., J. W. Holt, and N. Kaiser, 2015, Phys. Rev. C 92,015801.

Wellenhofer, C., J. W. Holt, N. Kaiser, and W. Weise, 2014,Phys. Rev. C 89, 064009.

Wen, D.-H., W. G. Newton, and B.-A. Li, 2012, Phys. Rev. C 85,025801.

Whittenbury, D., J. Carroll, A. Thomas, K. Tsushima, and J. Stone,2014, Phys. Rev. C 89, 065801.

Wienholtz, F., et al., 2013, Nature (London) 498, 346.Williams, R., and S. Koonin, 1985, Nucl. Phys. A 435, 844.Wiringa, R., R. Schiavilla, S. C. Pieper, and J. Carlson, 2014,Phys. Rev. C 89, 024305.

Wiringa, R. B., and S. C. Pieper, 2002, Phys. Rev. Lett. 89, 182501.Wiringa, R. B., R. Smith, and T. Ainsworth, 1984, Phys. Rev. C 29,1207.

Wiringa, R. B., V. Stoks, and R. Schiavilla, 1995, Phys. Rev. C 51,38.

Witten, E., 1984, Phys. Rev. D 30, 272.Wlazłowski, G., and P. Magierski, 2011, Phys. Rev. C 83, 012801.







http://dx.doi.org/10.1016/0375-9474(95)00161-S









http://dx.doi.org/10.1063/1.4795946




http://dx.doi.org/10.1134/S1063779615040061




http://dx.doi.org/10.1016/S0375-9474(99)00310-3







http://dx.doi.org/10.1088/0004-637X/728/2/95

http://dx.doi.org/10.1088/0004-637X/728/2/95

http://dx.doi.org/10.1088/1742-6596/342/1/012011

http://dx.doi.org/10.1088/1742-6596/342/1/012011



http://dx.doi.org/10.1016/0370-2693(70)90458-2


http://dx.doi.org/10.1016/0375-9474(92)90005-5

http://dx.doi.org/10.1086/529576




http://dx.doi.org/10.1209/0295-5075/94/11002





http://dx.doi.org/10.1016/S0375-9474(03)01313-7

http://dx.doi.org/10.1016/S0375-9474(03)01313-7




http://dx.doi.org/10.1016/0003-4916(74)90208-5



http://dx.doi.org/10.1088/2041-8205/789/2/L39


http://dx.doi.org/10.1088/1674-1137/36/12/003






http://dx.doi.org/10.1063/1.2713539


http://dx.doi.org/10.1016/S0375-9474(00)00197-4

http://dx.doi.org/10.1016/S0375-9474(00)00197-4




http://dx.doi.org/10.1016/S0375-9474(03)00893-5





http://dx.doi.org/10.1016/0370-2693(90)90938-3

http://dx.doi.org/10.1016/0550-3213(91)90231-L



http://dx.doi.org/10.1088/2041-8205/740/1/L14









http://dx.doi.org/10.1016/0375-9474(85)90191-5









Wlazłowski, G., J. Holt, S. Moroz, A. Bulgac, and K. Roche, 2014,Phys. Rev. Lett. 113, 182503.

Wolf, R., et al. (ISOLTRAP Collaboration), 2013, Phys. Rev. Lett.110, 041101.

Wolff, R. G., 1980, Diplomathesis, T. H. Darmstadt (unpublished).Wolter, H., et al., 2009, Prog. Part. Nucl. Phys. 62, 402.Woosley, S. E., and T. A. Weaver, 1995, Astrophys. J. Suppl. Ser.101, 181.

Xiao, Z., B.-A. Li, L.-W. Chen, G.-C. Yong, and M. Zhang, 2009,Phys. Rev. Lett. 102, 062502.

Xiao, Z.-G., et al., 2014, Eur. Phys. J. A 50, 37.Xie,W.-J., J. Su, L.Zhu, andF.-S.Zhang, 2013,Phys.Lett. B718, 1510.Xu, C., B.-A. Li, and L.-W. Chen, 2010, Phys. Rev. C 82, 054607.Xu, J., L.-W. Chen, and B.-A. Li, 2015, Phys. Rev. C 91, 014611.Yakovlev, D. G., and C. Pethick, 2004, Annu. Rev. Astron.Astrophys. 42, 169.

Yamamoto, Y., S.-i. Fujimoto, H. Nagakura, and S. Yamada, 2013,Astrophys. J. 771, 27.

Yang, B., Z.-P. Jin, X. Li, S. Covino, X.-Z. Zheng, K. Hotokezaka,Y.-Z. Fan, T. Piran, and D.-M. Wei, 2015, Nat. Commun. 6, 7323.

Yasutake, N., and K. Kashiwa, 2009, Phys. Rev. D 79, 043012.Yasutake, N., T. Maruyama, and T. Tatsumi, 2011, J. Phys. Conf. Ser.312, 042027.

Yasutake, N., T. Noda, H. Sotani, T. Maruyama, and T. Tatsumi,2013, in Recent Advances in Quarks Research (Nova SciencePublishers, New York), pp. 63–111.

Yong, G.-C., B.-A. Li, L.-W. Chen, and X.-C. Zhang, 2009,Phys. Rev. C 80, 044608.

Yudin, A., 2010, “Excluded volume approximation for supernovamatter,” in PoS(NIC XI)15.

Yudin, A. V., 2011, Astron. Lett. 37, 576.Yukawa, H., 1955, Prog. Theor. Phys. 1, 1.Zacchi, A., R. Stiele, and J. Schaffner-Bielich, 2015, Phys. Rev. D 92,045022.

Zdunik, J., 2000, Astron. Astrophys. 359, 311.Zdunik, J., and P. Haensel, 2013, Astron. Astrophys. 551, A61.Zenihiro, J., et al., 2010, Phys. Rev. C 82, 044611.Zhang, Y., M. Tsang, Z. Li, and H. Liu, 2014, Phys. Lett. B 732,186.

Zhang, Z., and L.-W. Chen, 2013, Phys. Lett. B 726, 234.Zhang, Z., and L.-W. Chen, 2015, Phys. Rev. C 92, 031301.Zhang, Z., and H. Shen, 2014, Astrophys. J. 788, 185.Zhao, P., Z. Li, J. Yao, and J. Meng, 2010, Phys. Rev. C 82,054319.

Zhou, X. R., G. F. Burgio, U. Lombardo, H.-J. Schulze, and W. Zuo,2004, Phys. Rev. C 69, 018801.

Zimanyi, J., and S. Moszkowski, 1990, Phys. Rev. C 42, 1416.Zuo, W., A. Lejeune, U. Lombardo, and J. Mathiot, 2002a,Nucl. Phys. A 706, 418.

Zuo, W., A. Lejeune, U. Lombardo, and J. Mathiot, 2002b, Eur. Phys.J. A 14, 469.







http://dx.doi.org/10.1086/192237

http://dx.doi.org/10.1086/192237






http://dx.doi.org/10.1146/annurev.astro.42.053102.134013

http://dx.doi.org/10.1146/annurev.astro.42.053102.134013

http://dx.doi.org/10.1088/0004-637X/771/1/27



http://dx.doi.org/10.1088/1742-6596/312/4/042027

http://dx.doi.org/10.1088/1742-6596/312/4/042027


http://dx.doi.org/10.1134/S0320010811080067




http://dx.doi.org/10.1051/0004-6361/201220697






http://dx.doi.org/10.1088/0004-637X/788/2/185





http://dx.doi.org/10.1016/S0375-9474(02)00750-9



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